%------------------------------------------------------------------------------
% File     : ITP214_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Hoare_Triple 00428_012651
% 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    : 0025_Hoare_Triple_00428_012651 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.1.0
% Syntax   : Number of formulae    : 12183 (2883 unt;2093 typ;   0 def)
%            Number of atoms       : 27552 (8003 equ)
%            Maximal formula atoms :   38 (   2 avg)
%            Number of connectives : 19795 (2333   ~; 335   |;1926   &)
%                                         (2246 <=>;12955  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   6 avg)
%            Maximal term depth    :   40 (   2 avg)
%            Number of types       :   13 (  12 usr)
%            Number of type conns  : 1913 (1540   >; 373   *;   0   +;   0  <<)
%            Number of predicates  :  231 ( 228 usr;   2 prp; 0-6 aty)
%            Number of functors    : 1853 (1853 usr;  68 con; 0-8 aty)
%            Number of variables   : 39837 (35704   !; 747   ?;39837   :)
%                                         (3386  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_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
%            2022-02-17 15:41:19.784
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
tff(ty_t_Heap__Time__Monad_OHeap,type,
    heap_Time_Heap: $tType > $tType ).

tff(ty_t_Code__Numeral_Onatural,type,
    code_natural: $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_Multiset_Omultiset,type,
    multiset: $tType > $tType ).

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

tff(ty_t_Sum__Type_Osum,type,
    sum_sum: ( $tType * $tType ) > $tType ).

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

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

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

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

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

tff(ty_t_Heap_Oref,type,
    ref: $tType > $tType ).

tff(ty_t_HOL_Obool,type,
    bool: $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 (2068)
tff(sy_cl_Lattices_Obounded__lattice,type,
    bounded_lattice: 
      !>[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_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_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_Orderings_Otop,type,
    top: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

tff(sy_cl_GCD_Osemiring__Gcd,type,
    semiring_Gcd: 
      !>[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_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_Groups_Omonoid__mult,type,
    monoid_mult: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_Lattices_Odistrib__lattice,type,
    distrib_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_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_Enum_Ofinite__distrib__lattice,type,
    finite8700451911770168679attice: 
      !>[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_Bit__Operations_Osemiring__bits,type,
    bit_semiring_bits: 
      !>[A: $tType] : $o ).

tff(sy_cl_Lattices_Obounded__lattice__top,type,
    bounded_lattice_top: 
      !>[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_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_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_Boolean__Algebras_Oboolean__algebra,type,
    boolea8198339166811842893lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__semiring__strict,type,
    linord8928482502909563296strict: 
      !>[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_Orderings_Ounbounded__dense__linorder,type,
    unboun7993243217541854897norder: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
    comple5582772986160207858norder: 
      !>[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_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_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_Euclidean__Division_Oeuclidean__ring__cancel,type,
    euclid8851590272496341667cancel: 
      !>[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_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_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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ct____,type,
    aTP_Lamp_ct: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cu____,type,
    aTP_Lamp_cu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cv____,type,
    aTP_Lamp_cv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cw____,type,
    aTP_Lamp_cw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cx____,type,
    aTP_Lamp_cx: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : ( 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] : fun(A,B) ).

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

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

tff(sy_c_ATP_058Lamp__du____,type,
    aTP_Lamp_du: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__fk____,type,
    aTP_Lamp_fk: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fo____,type,
    aTP_Lamp_fo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fp____,type,
    aTP_Lamp_fp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ge____,type,
    aTP_Lamp_ge: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gf____,type,
    aTP_Lamp_gf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gg____,type,
    aTP_Lamp_gg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gh____,type,
    aTP_Lamp_gh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

tff(sy_c_ATP_058Lamp__gl____,type,
    aTP_Lamp_gl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gm____,type,
    aTP_Lamp_gm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__go____,type,
    aTP_Lamp_go: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gp____,type,
    aTP_Lamp_gp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gq____,type,
    aTP_Lamp_gq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gr____,type,
    aTP_Lamp_gr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gs____,type,
    aTP_Lamp_gs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gt____,type,
    aTP_Lamp_gt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gu____,type,
    aTP_Lamp_gu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gv____,type,
    aTP_Lamp_gv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gw____,type,
    aTP_Lamp_gw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gx____,type,
    aTP_Lamp_gx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gy____,type,
    aTP_Lamp_gy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gz____,type,
    aTP_Lamp_gz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__kn____,type,
    aTP_Lamp_kn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ko____,type,
    aTP_Lamp_ko: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ld____,type,
    aTP_Lamp_ld: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__le____,type,
    aTP_Lamp_le: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

tff(sy_c_ATP_058Lamp__lj____,type,
    aTP_Lamp_lj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lk____,type,
    aTP_Lamp_lk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ll____,type,
    aTP_Lamp_ll: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lm____,type,
    aTP_Lamp_lm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ln____,type,
    aTP_Lamp_ln: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lo____,type,
    aTP_Lamp_lo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lp____,type,
    aTP_Lamp_lp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lq____,type,
    aTP_Lamp_lq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lr____,type,
    aTP_Lamp_lr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ls____,type,
    aTP_Lamp_ls: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lt____,type,
    aTP_Lamp_lt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lu____,type,
    aTP_Lamp_lu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lv____,type,
    aTP_Lamp_lv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lw____,type,
    aTP_Lamp_lw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lx____,type,
    aTP_Lamp_lx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ly____,type,
    aTP_Lamp_ly: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lz____,type,
    aTP_Lamp_lz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ma____,type,
    aTP_Lamp_ma: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mb____,type,
    aTP_Lamp_mb: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mk____,type,
    aTP_Lamp_mk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ml____,type,
    aTP_Lamp_ml: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mm____,type,
    aTP_Lamp_mm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mn____,type,
    aTP_Lamp_mn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mo____,type,
    aTP_Lamp_mo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mp____,type,
    aTP_Lamp_mp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mq____,type,
    aTP_Lamp_mq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mr____,type,
    aTP_Lamp_mr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ms____,type,
    aTP_Lamp_ms: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mt____,type,
    aTP_Lamp_mt: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__nq____,type,
    aTP_Lamp_nq: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__nz____,type,
    aTP_Lamp_nz: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__od____,type,
    aTP_Lamp_od: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ox____,type,
    aTP_Lamp_ox: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pa____,type,
    aTP_Lamp_pa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pb____,type,
    aTP_Lamp_pb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pc____,type,
    aTP_Lamp_pc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pd____,type,
    aTP_Lamp_pd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pe____,type,
    aTP_Lamp_pe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pf____,type,
    aTP_Lamp_pf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pg____,type,
    aTP_Lamp_pg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ph____,type,
    aTP_Lamp_ph: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pi____,type,
    aTP_Lamp_pi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pj____,type,
    aTP_Lamp_pj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pk____,type,
    aTP_Lamp_pk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pl____,type,
    aTP_Lamp_pl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pm____,type,
    aTP_Lamp_pm: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qm____,type,
    aTP_Lamp_qm: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qp____,type,
    aTP_Lamp_qp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qq____,type,
    aTP_Lamp_qq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qr____,type,
    aTP_Lamp_qr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qs____,type,
    aTP_Lamp_qs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qt____,type,
    aTP_Lamp_qt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qu____,type,
    aTP_Lamp_qu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qv____,type,
    aTP_Lamp_qv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qw____,type,
    aTP_Lamp_qw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qx____,type,
    aTP_Lamp_qx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qy____,type,
    aTP_Lamp_qy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qz____,type,
    aTP_Lamp_qz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ra____,type,
    aTP_Lamp_ra: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rb____,type,
    aTP_Lamp_rb: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rf____,type,
    aTP_Lamp_rf: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sd____,type,
    aTP_Lamp_sd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__se____,type,
    aTP_Lamp_se: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sf____,type,
    aTP_Lamp_sf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sg____,type,
    aTP_Lamp_sg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sh____,type,
    aTP_Lamp_sh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__si____,type,
    aTP_Lamp_si: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sj____,type,
    aTP_Lamp_sj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sk____,type,
    aTP_Lamp_sk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sl____,type,
    aTP_Lamp_sl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sm____,type,
    aTP_Lamp_sm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sn____,type,
    aTP_Lamp_sn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__so____,type,
    aTP_Lamp_so: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tk____,type,
    aTP_Lamp_tk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tl____,type,
    aTP_Lamp_tl: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__to____,type,
    aTP_Lamp_to: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tp____,type,
    aTP_Lamp_tp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tq____,type,
    aTP_Lamp_tq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tr____,type,
    aTP_Lamp_tr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ts____,type,
    aTP_Lamp_ts: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tt____,type,
    aTP_Lamp_tt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tu____,type,
    aTP_Lamp_tu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tv____,type,
    aTP_Lamp_tv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tw____,type,
    aTP_Lamp_tw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tx____,type,
    aTP_Lamp_tx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ty____,type,
    aTP_Lamp_ty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tz____,type,
    aTP_Lamp_tz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uc____,type,
    aTP_Lamp_uc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ud____,type,
    aTP_Lamp_ud: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ue____,type,
    aTP_Lamp_ue: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uf____,type,
    aTP_Lamp_uf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ug____,type,
    aTP_Lamp_ug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vm____,type,
    aTP_Lamp_vm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vn____,type,
    aTP_Lamp_vn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vo____,type,
    aTP_Lamp_vo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vp____,type,
    aTP_Lamp_vp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vq____,type,
    aTP_Lamp_vq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vr____,type,
    aTP_Lamp_vr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vs____,type,
    aTP_Lamp_vs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vt____,type,
    aTP_Lamp_vt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vu____,type,
    aTP_Lamp_vu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vv____,type,
    aTP_Lamp_vv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vw____,type,
    aTP_Lamp_vw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vx____,type,
    aTP_Lamp_vx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vy____,type,
    aTP_Lamp_vy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vz____,type,
    aTP_Lamp_vz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wa____,type,
    aTP_Lamp_wa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wb____,type,
    aTP_Lamp_wb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wc____,type,
    aTP_Lamp_wc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wd____,type,
    aTP_Lamp_wd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__we____,type,
    aTP_Lamp_we: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wf____,type,
    aTP_Lamp_wf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wg____,type,
    aTP_Lamp_wg: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wp____,type,
    aTP_Lamp_wp: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wu____,type,
    aTP_Lamp_wu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xl____,type,
    aTP_Lamp_xl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xn____,type,
    aTP_Lamp_xn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xp____,type,
    aTP_Lamp_xp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xq____,type,
    aTP_Lamp_xq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xr____,type,
    aTP_Lamp_xr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xs____,type,
    aTP_Lamp_xs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xt____,type,
    aTP_Lamp_xt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xu____,type,
    aTP_Lamp_xu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xv____,type,
    aTP_Lamp_xv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xw____,type,
    aTP_Lamp_xw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xx____,type,
    aTP_Lamp_xx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xy____,type,
    aTP_Lamp_xy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xz____,type,
    aTP_Lamp_xz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ya____,type,
    aTP_Lamp_ya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yb____,type,
    aTP_Lamp_yb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yc____,type,
    aTP_Lamp_yc: 
      !>[A: $tType,B: $tType] : 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Array__Time_Oalloc,type,
    array_alloc: 
      !>[A: $tType] : ( ( list(A) * heap_ext(product_unit) ) > product_prod(array(A),heap_ext(product_unit)) ) ).

tff(sy_c_Array__Time_Ofreeze,type,
    array_freeze: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_Array__Time_Oget,type,
    array_get: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > list(A) ) ).

tff(sy_c_Array__Time_Olen,type,
    array_len: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).

tff(sy_c_Array__Time_Olen_H,type,
    array_len2: 
      !>[B: $tType,A: $tType] : ( array(B) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Olength,type,
    array_length: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > 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_Opresent,type,
    array_present: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > $o ) ).

tff(sy_c_Array__Time_Oset,type,
    array_set: 
      !>[A: $tType] : ( ( array(A) * list(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

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_Array__Time_Oupd_H,type,
    array_upd2: 
      !>[A: $tType] : ( ( array(A) * code_integer * A ) > heap_Time_Heap(product_unit) ) ).

tff(sy_c_Array__Time_Oupdate,type,
    array_update: 
      !>[A: $tType] : ( ( array(A) * nat * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

tff(sy_c_Assertions_Oassn_OAbs__assn,type,
    abs_assn: fun(product_prod(heap_ext(product_unit),set(nat)),bool) > assn ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

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_Oin__range,type,
    in_range: fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

tff(sy_c_Assertions_Oin__range__rel,type,
    in_range_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).

tff(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

tff(sy_c_Assertions_Oone__assn__raw__rel,type,
    one_assn_raw_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).

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: bool > assn ).

tff(sy_c_Assertions_Opure__assn__raw,type,
    pure_assn_raw: 
      !>[A: $tType,B: $tType] : ( bool > fun(product_prod(A,set(B)),bool) ) ).

tff(sy_c_Assertions_Opure__assn__raw__rel,type,
    pure_assn_raw_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(bool,product_prod(A,set(B))),fun(product_prod(bool,product_prod(A,set(B))),bool)) ).

tff(sy_c_Assertions_OrelH,type,
    relH: ( set(nat) * heap_ext(product_unit) * heap_ext(product_unit) ) > $o ).

tff(sy_c_Assertions_Osnga__assn,type,
    snga_assn: 
      !>[A: $tType] : ( ( array(A) * list(A) ) > assn ) ).

tff(sy_c_Assertions_Osnga__assn__raw,type,
    snga_assn_raw: 
      !>[A: $tType] : ( ( array(A) * list(A) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) ).

tff(sy_c_Assertions_Osnga__assn__raw__rel,type,
    snga_assn_raw_rel: 
      !>[A: $tType] : fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).

tff(sy_c_Assertions_Osngr__assn,type,
    sngr_assn: 
      !>[A: $tType] : ( ( ref(A) * A ) > assn ) ).

tff(sy_c_Assertions_Osngr__assn__raw,type,
    sngr_assn_raw: 
      !>[A: $tType] : ( ( ref(A) * A ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) ).

tff(sy_c_Assertions_Osngr__assn__raw__rel,type,
    sngr_assn_raw_rel: 
      !>[A: $tType] : fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool)) ).

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).

tff(sy_c_Assertions_Owand__assn,type,
    wand_assn: ( assn * assn ) > assn ).

tff(sy_c_Assertions_Owand__raw,type,
    wand_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) ) > fun(product_prod(heap_ext(product_unit),set(nat)),bool) ).

tff(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).

tff(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
    bNF_Cardinal_Csum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
    bNF_Ca4139267488887388095finite: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
    bNF_Cardinal_cprod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
    bNF_Cardinal_csum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
    bNF_Ca8387033319878233205ardSuc: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
    bNF_Ca6860139660246222851ard_of: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
    bNF_Ca8970107618336181345der_on: 
      !>[A: $tType] : ( set(A) > fun(set(product_prod(A,A)),bool) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
    bNF_Ca7293521722713021262ofinal: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
    bNF_Ca6246979054910435723ardSuc: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(set(A),set(A))),bool) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
    bNF_Ca7133664381575040944arCard: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
    bNF_Ca3754400796208372196lChain: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).

tff(sy_c_BNF__Def_OGr,type,
    bNF_Gr: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Def_OGrp,type,
    bNF_Grp: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,bool)) ) ).

tff(sy_c_BNF__Def_Ocollect,type,
    bNF_collect: 
      !>[B: $tType,A: $tType] : ( set(fun(B,set(A))) > fun(B,set(A)) ) ).

tff(sy_c_BNF__Def_Oconvol,type,
    bNF_convol: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,B) * fun(A,C) ) > fun(A,product_prod(B,C)) ) ).

tff(sy_c_BNF__Def_Oeq__onp,type,
    bNF_eq_onp: 
      !>[A: $tType] : ( fun(A,bool) > fun(A,fun(A,bool)) ) ).

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).

tff(sy_c_BNF__Def_Orel__set,type,
    bNF_rel_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(set(A),fun(set(B),bool)) ) ).

tff(sy_c_BNF__Def_Ovimage2p,type,
    bNF_vimage2p: 
      !>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( fun(A,D) * fun(B,E) * fun(D,fun(E,C)) ) > fun(A,fun(B,C)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift,type,
    bNF_Greatest_Shift: 
      !>[A: $tType] : ( ( set(list(A)) * A ) > set(list(A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
    bNF_Greatest_Succ: 
      !>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
    bNF_Gr5436034075474128252omCard: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * B ) > A ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
    bNF_Greatest_image2: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
    bNF_Greatest_image2p: 
      !>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( fun(C,A) * fun(D,B) * fun(C,fun(D,bool)) ) > fun(A,fun(B,bool)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
    bNF_Gr4221423524335903396lImage: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
    bNF_Gr7122648621184425601vImage: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
    bNF_Greatest_shift: 
      !>[A: $tType,B: $tType] : ( ( fun(list(A),B) * A * list(A) ) > B ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
    bNF_Greatest_toCard: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(A,B) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
    bNF_Gr1419584066657907630d_pred: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(fun(A,B),bool) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
    bNF_Greatest_univ: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > A ) ).

tff(sy_c_BNF__Wellorder__Constructions_OFunc,type,
    bNF_Wellorder_Func: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
    bNF_We4925052301507509544nc_map: 
      !>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,A)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
    bNF_Wellorder_bsqr: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(product_prod(A,A),product_prod(A,A))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
    bNF_Wellorder_curr: 
      !>[A: $tType,B: $tType,C: $tType] : ( set(A) > fun(fun(product_prod(A,B),C),fun(A,fun(B,C))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
    bNF_We413866401316099525erIncl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
    bNF_Wellorder_ordIso: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
    bNF_Wellorder_ordLeq: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
    bNF_We4044943003108391690rdLess: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
    bNF_We8469521843155493636filter: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(A,A)),set(A)) ) ).

tff(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
    bNF_Wellorder_compat: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Embedding_Oembed,type,
    bNF_Wellorder_embed: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) ) > fun(fun(A,A2),bool) ) ).

tff(sy_c_BNF__Wellorder__Embedding_OembedS,type,
    bNF_Wellorder_embedS: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Embedding_Oiso,type,
    bNF_Wellorder_iso: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
    bNF_Wellorder_wo_rel: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
    bNF_We4791949203932849705sMinim: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > fun(A,bool) ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
    bNF_We1388413361240627857o_max2: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
    bNF_We6954850376910717587_minim: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
    bNF_Wellorder_wo_suc: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

tff(sy_c_Basic__BNFs_Ofsts,type,
    basic_fsts: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(A) ) ).

tff(sy_c_Basic__BNFs_Opred__fun,type,
    basic_pred_fun: 
      !>[A: $tType,B: $tType] : ( fun(A,bool) > fun(fun(B,bool),fun(fun(A,B),bool)) ) ).

tff(sy_c_Basic__BNFs_Opred__prod,type,
    basic_pred_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) ) > fun(product_prod(A,B),bool) ) ).

tff(sy_c_Basic__BNFs_Orel__prod,type,
    basic_rel_prod: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,bool)) * fun(C,fun(D,bool)) ) > fun(product_prod(A,C),fun(product_prod(B,D),bool)) ) ).

tff(sy_c_Basic__BNFs_Osnds,type,
    basic_snds: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).

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__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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] : ( 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] : ( ( A * 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] : ( ( A * 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] : ( ( A * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,bool) ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > bool ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * num ) > option(num) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
    boolea2506097494486148201lgebra: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,bool) ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: fun(code_integer,int) ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun(int,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: fun(code_natural,nat) ).

tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: fun(nat,code_natural) ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Numeral_Osub,type,
    code_sub: fun(num,fun(num,code_integer)) ).

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
    complete_Sup_Sup: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
    comple1908693960933563346ssible: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,bool) ) > $o ) ).

tff(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
    comple7512665784863727008ratesp: 
      !>[A: $tType] : ( fun(A,A) > fun(A,bool) ) ).

tff(sy_c_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
    condit8047198070973881523_above: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),bool) ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
    condit8119078960628432327_below: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),bool) ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
    condit941137186595557371_above: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
    condit1013018076250108175_below: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_Equiv__Relations_Ocongruent,type,
    equiv_congruent: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).

tff(sy_c_Equiv__Relations_Ocongruent2,type,
    equiv_congruent2: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).

tff(sy_c_Equiv__Relations_Oequiv,type,
    equiv_equiv: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Equiv__Relations_Oproj,type,
    equiv_proj: 
      !>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,set(A)) ) ).

tff(sy_c_Equiv__Relations_Oquotient,type,
    equiv_quotient: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > set(set(A)) ) ).

tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
    euclid6346220572633701492n_size: 
      !>[A: $tType] : ( A > nat ) ).

tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
    euclid7384307370059645450egment: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofilter_OAbs__filter,type,
    abs_filter: 
      !>[A: $tType] : ( fun(fun(A,bool),bool) > filter(A) ) ).

tff(sy_c_Filter_Ofiltercomap,type,
    filtercomap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofiltermap,type,
    filtermap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Filter_Ofinite__subsets__at__top,type,
    finite5375528669736107172at_top: 
      !>[A: $tType] : ( set(A) > filter(set(A)) ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Filter_Oprincipal,type,
    principal: 
      !>[A: $tType] : ( set(A) > filter(A) ) ).

tff(sy_c_Filter_Oprod__filter,type,
    prod_filter: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * filter(B) ) > filter(product_prod(A,B)) ) ).

tff(sy_c_Filter_Orel__filter,type,
    rel_filter: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(filter(A),fun(filter(B),bool)) ) ).

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,type,
    finite6289374366891150609ommute: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

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),bool) ).

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_Finite__Set_Ofold__graph,type,
    finite_fold_graph: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > fun(B,bool) ) ).

tff(sy_c_Finite__Set_Ofolding__idem__on,type,
    finite1890593828518410140dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on,type,
    finite_folding_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on_OF,type,
    finite_folding_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(set(A),B) ) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(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_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).

tff(sy_c_Fun_Ooverride__on,type,
    override_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).

tff(sy_c_Fun_Ostrict__mono__on,type,
    strict_mono_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).

tff(sy_c_Fun__Def_Omax__strict,type,
    fun_max_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omax__weak,type,
    fun_max_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__strict,type,
    fun_min_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__weak,type,
    fun_min_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Oreduction__pair,type,
    fun_reduction_pair: 
      !>[A: $tType] : ( product_prod(set(product_prod(A,A)),set(product_prod(A,A))) > $o ) ).

tff(sy_c_Fun__Def_Orp__inv__image,type,
    fun_rp_inv_image: 
      !>[A: $tType,B: $tType] : fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
    semiring_gcd_Gcd_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

tff(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Otimes__class_Otimes,type,
    times_times: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Ouminus__class_Ouminus,type,
    uminus_uminus: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ozero__class_Ozero,type,
    zero_zero: 
      !>[A: $tType] : A ).

tff(sy_c_Groups__Big_Ocomm__monoid__add_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(B,A) * set(B) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > A ) ).

tff(sy_c_Groups__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(fun(B,A),fun(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] : fun(list(A),A) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_HOL_OUniq,type,
    uniq: 
      !>[A: $tType] : ( fun(A,bool) > $o ) ).

tff(sy_c_HOL_Odefault__class_Odefault,type,
    default_default: 
      !>[A: $tType] : A ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Heap_Oaddr__of__array,type,
    addr_of_array: 
      !>[A: $tType] : ( array(A) > nat ) ).

tff(sy_c_Heap_Oaddr__of__ref,type,
    addr_of_ref: 
      !>[A: $tType] : ( ref(A) > nat ) ).

tff(sy_c_Heap_Oarray_OArray,type,
    array2: 
      !>[A: $tType] : ( nat > array(A) ) ).

tff(sy_c_Heap_Oheap_Olim,type,
    lim: 
      !>[Z: $tType] : ( heap_ext(Z) > nat ) ).

tff(sy_c_Heap_Oheap_Olim__update,type,
    lim_update: 
      !>[Z: $tType] : ( ( fun(nat,nat) * heap_ext(Z) ) > heap_ext(Z) ) ).

tff(sy_c_Heap_Oref_ORef,type,
    ref2: 
      !>[A: $tType] : ( nat > ref(A) ) ).

tff(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
    heap_Time_Heap2: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
    heap_Time_size_Heap: 
      !>[A: $tType] : ( ( fun(A,nat) * heap_Time_Heap(A) ) > nat ) ).

tff(sy_c_Heap__Time__Monad_OHeap__lub,type,
    heap_Time_Heap_lub: 
      !>[A: $tType] : ( set(heap_Time_Heap(A)) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oassert,type,
    heap_Time_assert: 
      !>[A: $tType] : ( ( fun(A,bool) * A ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Obind,type,
    heap_Time_bind: 
      !>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).

tff(sy_c_Heap__Time__Monad_Oeffect,type,
    heap_Time_effect: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A * nat ) > $o ) ).

tff(sy_c_Heap__Time__Monad_Oexecute,type,
    heap_Time_execute: 
      !>[A: $tType] : ( heap_Time_Heap(A) > fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).

tff(sy_c_Heap__Time__Monad_Oguard,type,
    heap_Time_guard: 
      !>[A: $tType] : ( ( fun(heap_ext(product_unit),bool) * fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oheap,type,
    heap_Time_heap: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Olift,type,
    heap_Time_lift: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(A,heap_Time_Heap(B)) ) ).

tff(sy_c_Heap__Time__Monad_Oraise,type,
    heap_Time_raise: 
      !>[A: $tType] : ( list(char) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oreturn,type,
    heap_Time_return: 
      !>[A: $tType] : fun(A,heap_Time_Heap(A)) ).

tff(sy_c_Heap__Time__Monad_Osuccess,type,
    heap_Time_success: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).

tff(sy_c_Heap__Time__Monad_Otap,type,
    heap_Time_tap: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),A) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_OtimeFrame,type,
    heap_Time_timeFrame: 
      !>[A: $tType] : ( ( nat * option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ).

tff(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
    heap_T5500966940807335491me_rel: 
      !>[A: $tType] : fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool)) ).

tff(sy_c_Heap__Time__Monad_Oureturn,type,
    heap_Time_ureturn: 
      !>[A: $tType] : fun(A,heap_Time_Heap(A)) ).

tff(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_Time_Heap(product_unit) ).

tff(sy_c_Hoare__Triple_Ohoare__triple,type,
    hoare_hoare_triple: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) ) > $o ) ).

tff(sy_c_Hoare__Triple_Onew__addrs,type,
    hoare_new_addrs: ( heap_ext(product_unit) * set(nat) * heap_ext(product_unit) ) > set(nat) ).

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : ( ( bool * A * A ) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
    complete_lattice_gfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
    complete_lattice_lfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( ( set(A) * nat ) > A ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_Int_Onat,type,
    nat2: fun(int,nat) ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,bool)) ).

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_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,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > 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_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
    lattices_ord_arg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
    lattic1883929316492267755max_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
    lattices_ord_arg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).

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_Lattices__Big_Oord__class_Ois__arg__max,type,
    lattic501386751176901750rg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > fun(B,bool) ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
    lattic501386751177426532rg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > fun(B,bool) ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
    lattic8678736583308907530nf_fin: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
    lattic4895041142388067077er_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
    lattic4630905495605216202up_fin: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lifting_OQuotient,type,
    quotient: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) * fun(A,fun(B,bool)) ) > $o ) ).

tff(sy_c_Lifting_Orel__pred__comp,type,
    rel_pred_comp: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,bool) * A ) > $o ) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).

tff(sy_c_List_Oarg__min__list,type,
    arg_min_list: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),bool)) ).

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_Ocoset,type,
    coset: 
      !>[A: $tType] : ( list(A) > set(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,bool) * 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,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofold,type,
    fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * set(B) * fun(B,A) ) > $o ) ).

tff(sy_c_List_Ofoldl,type,
    foldl: 
      !>[B: $tType,A: $tType] : ( fun(B,fun(A,B)) > fun(B,fun(list(A),B)) ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Ogen__length,type,
    gen_length: 
      !>[A: $tType] : ( ( nat * list(A) ) > nat ) ).

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(nat,set(product_prod(list(A),list(A)))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexordp,type,
    lexordp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * set(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
    linord329482645794927042rt_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

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_Olinorder__class_Ostable__sort__key,type,
    linord3483353639454293061rt_key: 
      !>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(list(A),list(Aa)) ) ).

tff(sy_c_List_Olist_Orec__list,type,
    rec_list: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,fun(list(A),fun(C,C))) ) > fun(list(A),C) ) ).

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_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olistrel1p,type,
    listrel1p: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

tff(sy_c_List_Olists,type,
    lists: 
      !>[A: $tType] : ( 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_Omap__project,type,
    map_project: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > set(B) ) ).

tff(sy_c_List_Omeasures,type,
    measures: 
      !>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).

tff(sy_c_List_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oord_Olexordp,type,
    lexordp2: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),fun(list(A),bool)) ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > product_prod(list(A),list(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_Oremdups,type,
    remdups: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Oremdups__adj,type,
    remdups_adj: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * 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] : ( list(A) > list(A) ) ).

tff(sy_c_List_Orotate,type,
    rotate: 
      !>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( 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_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles__rel,type,
    shuffles_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * 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,bool) * 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)),bool)) ).

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

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__add,type,
    map_add: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__comp,type,
    map_comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,option(C)) * fun(A,option(B)) * A ) > option(C) ) ).

tff(sy_c_Map_Omap__le,type,
    map_le: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > $o ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(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_OCODE__ABORT,type,
    cODE_ABORT: 
      !>[A: $tType] : ( fun(product_unit,A) > A ) ).

tff(sy_c_Misc_OEps__Opt,type,
    eps_Opt: 
      !>[A: $tType] : ( fun(A,bool) > option(A) ) ).

tff(sy_c_Misc_Obijective,type,
    bijective: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

tff(sy_c_Misc_Obrk__rel,type,
    brk_rel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(product_prod(bool,A),product_prod(bool,B))) ) ).

tff(sy_c_Misc_Odflt__None__set,type,
    dflt_None_set: 
      !>[A: $tType] : ( set(A) > option(set(A)) ) ).

tff(sy_c_Misc_Ofilter__rev,type,
    filter_rev: 
      !>[A: $tType] : fun(fun(A,bool),fun(list(A),list(A))) ).

tff(sy_c_Misc_Ofilter__rev__aux,type,
    filter_rev_aux: 
      !>[A: $tType] : ( list(A) > fun(fun(A,bool),fun(list(A),list(A))) ) ).

tff(sy_c_Misc_Ofun__of__rel,type,
    fun_of_rel: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > A ) ).

tff(sy_c_Misc_Oinv__on,type,
    inv_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > fun(B,A) ) ).

tff(sy_c_Misc_Olist__collect__set,type,
    list_collect_set: 
      !>[B: $tType,A: $tType] : ( ( fun(B,set(A)) * list(B) ) > set(A) ) ).

tff(sy_c_Misc_Omap__mmupd,type,
    map_mmupd: 
      !>[B: $tType,A: $tType] : ( ( fun(B,option(A)) * set(B) * A ) > fun(B,option(A)) ) ).

tff(sy_c_Misc_Omap__to__set,type,
    map_to_set: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

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__list__rel,type,
    merge_list_rel: 
      !>[A: $tType] : fun(product_prod(list(list(A)),list(list(A))),fun(product_prod(list(list(A)),list(list(A))),bool)) ).

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)),bool)) ).

tff(sy_c_Misc_Omergesort,type,
    mergesort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Misc_Omergesort__by__rel,type,
    mergesort_by_rel: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge,type,
    merges9089515139780605204_merge: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
    merges2244889521215249637ge_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).

tff(sy_c_Misc_Omergesort__by__rel__rel,type,
    mergesort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).

tff(sy_c_Misc_Omergesort__by__rel__split,type,
    merges295452479951948502_split: 
      !>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__split__rel,type,
    merges7066485432131860899it_rel: 
      !>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),bool)) ).

tff(sy_c_Misc_Omergesort__remdups,type,
    mergesort_remdups: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Misc_Opairself,type,
    pairself: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(product_prod(A,A),product_prod(B,B)) ) ).

tff(sy_c_Misc_Opairself__rel,type,
    pairself_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),bool)) ).

tff(sy_c_Misc_Opartition__rev,type,
    partition_rev: 
      !>[A: $tType] : ( ( fun(A,bool) * product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Opartition__rev__rel,type,
    partition_rev_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),fun(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool)) ).

tff(sy_c_Misc_Oquicksort__by__rel,type,
    quicksort_by_rel: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Oquicksort__by__rel__rel,type,
    quicksort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),bool) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_Orel__restrict,type,
    rel_restrict: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Misc_Oremove__rev,type,
    remove_rev: 
      !>[A: $tType] : ( A > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Orevg,type,
    revg: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Orevg__rel,type,
    revg_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

tff(sy_c_Misc_Oset__to__map,type,
    set_to_map: 
      !>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,option(A)) ) ).

tff(sy_c_Misc_Oslice,type,
    slice: 
      !>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Othe__default,type,
    the_default: 
      !>[A: $tType] : ( ( A * option(A) ) > A ) ).

tff(sy_c_Misc_Ouncurry,type,
    uncurry: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

tff(sy_c_Misc_Ozipf,type,
    zipf: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * list(A) * list(B) ) > list(C) ) ).

tff(sy_c_Misc_Ozipf__rel,type,
    zipf_rel: 
      !>[A: $tType,B: $tType,C: $tType] : fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool)) ).

tff(sy_c_Multiset_Oadd__mset,type,
    add_mset: 
      !>[A: $tType] : fun(A,fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Ocomm__monoid__add_Osum__mset,type,
    comm_monoid_sum_mset: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * multiset(A) ) > A ) ).

tff(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
    comm_m7189776963980413722m_mset: 
      !>[A: $tType] : ( multiset(A) > A ) ).

tff(sy_c_Multiset_Ocr__multiset,type,
    cr_multiset: 
      !>[A: $tType] : fun(fun(A,nat),fun(multiset(A),bool)) ).

tff(sy_c_Multiset_Ofilter__mset,type,
    filter_mset: 
      !>[A: $tType] : fun(fun(A,bool),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Ofold__mset,type,
    fold_mset: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * multiset(A) ) > B ) ).

tff(sy_c_Multiset_Oimage__mset,type,
    image_mset: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(multiset(A),multiset(B)) ) ).

tff(sy_c_Multiset_Ointer__mset,type,
    inter_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).

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_Olinorder__class_Osorted__list__of__multiset,type,
    linord6283353356039996273ltiset: 
      !>[A: $tType] : ( multiset(A) > list(A) ) ).

tff(sy_c_Multiset_Oms__strict,type,
    ms_strict: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).

tff(sy_c_Multiset_Oms__weak,type,
    ms_weak: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).

tff(sy_c_Multiset_Omset,type,
    mset: 
      !>[A: $tType] : ( list(A) > multiset(A) ) ).

tff(sy_c_Multiset_Omset__set,type,
    mset_set: 
      !>[B: $tType] : ( set(B) > multiset(B) ) ).

tff(sy_c_Multiset_Omult,type,
    mult: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).

tff(sy_c_Multiset_Omult1,type,
    mult1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).

tff(sy_c_Multiset_Omulteqp__code,type,
    multeqp_code: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(multiset(A),fun(multiset(A),bool)) ) ).

tff(sy_c_Multiset_Omultiset_OAbs__multiset,type,
    abs_multiset: 
      !>[A: $tType] : fun(fun(A,nat),multiset(A)) ).

tff(sy_c_Multiset_Omultiset_Ocount,type,
    count: 
      !>[A: $tType] : fun(multiset(A),fun(A,nat)) ).

tff(sy_c_Multiset_Omultp,type,
    multp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(multiset(A),fun(multiset(A),bool)) ) ).

tff(sy_c_Multiset_Omultp__code,type,
    multp_code: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * multiset(A) * multiset(A) ) > $o ) ).

tff(sy_c_Multiset_Opcr__multiset,type,
    pcr_multiset: 
      !>[C: $tType,B: $tType] : ( fun(C,fun(B,bool)) > fun(fun(C,nat),fun(multiset(B),bool)) ) ).

tff(sy_c_Multiset_Opw__leq,type,
    pw_leq: ( multiset(product_prod(nat,nat)) * multiset(product_prod(nat,nat)) ) > $o ).

tff(sy_c_Multiset_Orel__mset,type,
    rel_mset: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(multiset(A),fun(multiset(B),bool)) ) ).

tff(sy_c_Multiset_Orepeat__mset,type,
    repeat_mset: 
      !>[A: $tType] : fun(nat,fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Oreplicate__mset,type,
    replicate_mset: 
      !>[A: $tType] : ( ( nat * A ) > multiset(A) ) ).

tff(sy_c_Multiset_Oset__mset,type,
    set_mset: 
      !>[A: $tType] : fun(multiset(A),set(A)) ).

tff(sy_c_Multiset_Osize__multiset,type,
    size_multiset: 
      !>[A: $tType] : ( fun(A,nat) > fun(multiset(A),nat) ) ).

tff(sy_c_Multiset_Osubset__eq__mset__impl,type,
    subset_eq_mset_impl: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > option(bool) ) ).

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)),bool)) ).

tff(sy_c_Multiset_Osubset__mset,type,
    subset_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),bool)) ).

tff(sy_c_Multiset_Osubseteq__mset,type,
    subseteq_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),bool)) ).

tff(sy_c_Multiset_Ounion__mset,type,
    union_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),multiset(A))) ).

tff(sy_c_Multiset_Owcount,type,
    wcount: 
      !>[A: $tType] : ( ( fun(A,nat) * multiset(A) * A ) > nat ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,bool) ) ).

tff(sy_c_Nat_Osemiring__1__class_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(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__decode,type,
    nat_list_decode: nat > list(nat) ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: nat > product_prod(nat,nat) ).

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),bool)) ).

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: set(nat) > nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

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_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun(num,num) ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one: 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_Num_Osqr,type,
    sqr: num > num ).

tff(sy_c_Option_Ocombine__options,type,
    combine_options: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * option(A) * option(A) ) > option(A) ) ).

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_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).

tff(sy_c_Option_Ooption_Orec__option,type,
    rec_option: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,C) ) > fun(option(A),C) ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( fun(A,nat) > fun(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_Order__Relation_OAbove,type,
    order_Above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OAboveS,type,
    order_AboveS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnder,type,
    order_Under: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnderS,type,
    order_UnderS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_Oabove,type,
    order_above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Olinear__order__on,type,
    order_679001287576687338der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Oofilter,type,
    order_ofilter: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).

tff(sy_c_Order__Relation_Opartial__order__on,type,
    order_7125193373082350890der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Opreorder__on,type,
    order_preorder_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Orelation__of,type,
    order_relation_of: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Order__Relation_Ounder,type,
    order_under: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(A,set(A)) ) ).

tff(sy_c_Order__Relation_OunderS,type,
    order_underS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Owell__order__on,type,
    order_well_order_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord_OLeast,type,
    least: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,bool),A) ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

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_Oorder_OGreatest,type,
    greatest: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,bool),A) ) ).

tff(sy_c_Orderings_Oorder_Omono,type,
    mono: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,B),bool) ) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),bool) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A ) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Partial__Function_Oflat__lub,type,
    partial_flat_lub: 
      !>[A: $tType] : ( ( A * set(A) ) > A ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Predicate_Oiterate__upto__rel,type,
    iterate_upto_rel: 
      !>[A: $tType] : fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool)) ).

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_OSigma,type,
    product_Sigma: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_OUnity,type,
    product_Unity: product_unit ).

tff(sy_c_Product__Type_Oapfst,type,
    product_apfst: 
      !>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).

tff(sy_c_Product__Type_Ocurry,type,
    product_curry: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(product_prod(A,B),C),fun(A,fun(B,C))) ).

tff(sy_c_Product__Type_Ointernal__case__prod,type,
    produc5280177257484947105e_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Omap__prod,type,
    product_map_prod: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
    product_rec_set_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
    product_rec_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > T ) ).

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_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Product__Type_Oprod_Oswap,type,
    product_swap: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).

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_Product__Type_Oscomp,type,
    product_scomp: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).

tff(sy_c_Product__Type_Ounit_OAbs__unit,type,
    product_Abs_unit: fun(bool,product_unit) ).

tff(sy_c_Product__Type_Ounit_ORep__unit,type,
    product_Rep_unit: fun(product_unit,bool) ).

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),bool)) ).

tff(sy_c_Random_Oinc__shift,type,
    inc_shift: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Oiterate,type,
    iterate: 
      !>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).

tff(sy_c_Random_Oiterate__rel,type,
    iterate_rel: 
      !>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool)) ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Olog__rel,type,
    log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),bool)) ).

tff(sy_c_Random_Ominus__shift,type,
    minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Onext,type,
    next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Opick,type,
    pick: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(code_natural,A) ) ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Oselect,type,
    select: 
      !>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Oselect__weight,type,
    select_weight: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Osplit__seed,type,
    split_seed: product_prod(code_natural,code_natural) > product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun(product_prod(int,int),rat) ).

tff(sy_c_Rat_OFract,type,
    fract: fun(int,fun(int,rat)) ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ocr__rat,type,
    cr_rat: ( product_prod(int,int) * rat ) > $o ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : fun(rat,A) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,bool)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,bool) ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Rat_Orat_OAbs__rat,type,
    abs_rat: fun(set(product_prod(int,int)),rat) ).

tff(sy_c_Rat_Orat_ORep__rat,type,
    rep_rat: fun(rat,set(product_prod(int,int))) ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Ref__Time_Oalloc,type,
    ref_alloc: 
      !>[A: $tType] : ( ( A * heap_ext(product_unit) ) > product_prod(ref(A),heap_ext(product_unit)) ) ).

tff(sy_c_Ref__Time_Ochange,type,
    ref_change: 
      !>[A: $tType] : ( ( fun(A,A) * ref(A) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Ref__Time_Oget,type,
    ref_get: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > A ) ).

tff(sy_c_Ref__Time_Olookup,type,
    ref_lookup: 
      !>[A: $tType] : ( ref(A) > heap_Time_Heap(A) ) ).

tff(sy_c_Ref__Time_Opresent,type,
    ref_present: 
      !>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > $o ) ).

tff(sy_c_Ref__Time_Oref,type,
    ref_ref: 
      !>[A: $tType] : ( A > heap_Time_Heap(ref(A)) ) ).

tff(sy_c_Ref__Time_Oset,type,
    ref_set: 
      !>[A: $tType] : ( ( ref(A) * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

tff(sy_c_Ref__Time_Oupdate,type,
    ref_update: 
      !>[A: $tType] : ( ( ref(A) * A ) > heap_Time_Heap(product_unit) ) ).

tff(sy_c_Relation_ODomainp,type,
    domainp: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(A,bool)) ).

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

tff(sy_c_Relation_OId__on,type,
    id_on: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_OImage,type,
    image: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > fun(set(A),set(B)) ) ).

tff(sy_c_Relation_Oantisym,type,
    antisym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oasymp,type,
    asymp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Oconverse,type,
    converse: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).

tff(sy_c_Relation_Oconversep,type,
    conversep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,fun(A,bool)) ) ).

tff(sy_c_Relation_Oinv__image,type,
    inv_image: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oirreflp,type,
    irreflp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Orefl__on,type,
    refl_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

tff(sy_c_Relation_Orelcompp,type,
    relcompp: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool)))) ).

tff(sy_c_Relation_Osingle__valued,type,
    single_valued: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

tff(sy_c_Relation_Osingle__valuedp,type,
    single_valuedp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Relation_Osym,type,
    sym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Otrans,type,
    trans: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Otransp,type,
    transp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
    algebr8660921524188924756oprime: 
      !>[A: $tType] : ( ( 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] : fun(A,fun(A,bool)) ).

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(bool,A) ).

tff(sy_c_Set_OBall,type,
    ball: 
      !>[A: $tType] : fun(set(A),fun(fun(A,bool),bool)) ).

tff(sy_c_Set_OBex,type,
    bex: 
      !>[A: $tType] : fun(set(A),fun(fun(A,bool),bool)) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,bool),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : fun(set(A),fun(set(A),bool)) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,bool) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ois__empty,type,
    is_empty: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Opairwise,type,
    pairwise: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : ( ( A * set(A) ) > set(A) ) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set_Ovimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),fun(set(B),set(A))) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_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))),bool)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : ( A > char ) ).

tff(sy_c_Sum__Type_OPlus,type,
    sum_Plus: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).

tff(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
    syntax7388354845996824322omatch: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_Transfer_Obi__total,type,
    bi_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Obi__unique,type,
    bi_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oleft__total,type,
    left_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oleft__unique,type,
    left_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oright__total,type,
    right_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Oright__unique,type,
    right_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transfer_Otransfer__bforall,type,
    transfer_bforall: 
      !>[A: $tType] : ( ( fun(A,bool) * fun(A,bool) ) > $o ) ).

tff(sy_c_Transitive__Closure_Oacyclic,type,
    transitive_acyclic: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(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_Typedef_Otype__definition,type,
    type_definition: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,bool) ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Olex__prod,type,
    lex_prod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),fun(set(A),bool)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Zorn_OChains,type,
    chains: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).

tff(sy_c_Zorn_Ochain__subset,type,
    chain_subset: 
      !>[A: $tType] : ( set(set(A)) > $o ) ).

tff(sy_c_Zorn_Ochains,type,
    chains2: 
      !>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).

tff(sy_c_Zorn_Oinit__seg__of,type,
    init_seg_of: 
      !>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).

tff(sy_c_Zorn_Opred__on_Ochain,type,
    pred_chain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) ) > fun(set(A),bool) ) ).

tff(sy_c_Zorn_Opred__on_Omaxchain,type,
    pred_maxchain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_Zorn_Opred__on_Osuc,type,
    pred_suc: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > set(A) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( A > fun(set(A),bool) ) ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_P,type,
    p: assn ).

tff(sy_v_Q,type,
    q: fun(b,assn) ).

tff(sy_v_R,type,
    r: fun(a,assn) ).

tff(sy_v_as____,type,
    as: set(nat) ).

tff(sy_v_f,type,
    f: heap_Time_Heap(a) ).

tff(sy_v_g,type,
    g: a > heap_Time_Heap(b) ).

tff(sy_v_h_H_H____,type,
    h: heap_ext(product_unit) ).

tff(sy_v_h_H____,type,
    h2: heap_ext(product_unit) ).

tff(sy_v_h____,type,
    h3: heap_ext(product_unit) ).

tff(sy_v_rf____,type,
    rf: a ).

tff(sy_v_rg____,type,
    rg: b ).

tff(sy_v_t_H_H____,type,
    t: nat ).

tff(sy_v_t_H____,type,
    t2: nat ).

% Relevant facts (9410)
tff(fact_0__092_060open_062new__addrs_Ah_H_A_Inew__addrs_Ah_Aas_Ah_H_J_Ah_H_H_A_061_Anew__addrs_Ah_Aas_Ah_H_H_092_060close_062,axiom,
    hoare_new_addrs(h2,hoare_new_addrs(h3,as,h2),h) = hoare_new_addrs(h3,as,h) ).

% \<open>new_addrs h' (new_addrs h as h') h'' = new_addrs h as h''\<close>
tff(fact_1__092_060open_062_Ih_M_Aas_J_A_092_060Turnstile_062_AP_092_060close_062,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(p),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),h3),as))) ).

% \<open>(h, as) \<Turnstile> P\<close>
tff(fact_2_new__addr__refl,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] : hoare_new_addrs(H,As,H) = As ).

% new_addr_refl
tff(fact_3_POST__G,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(b,assn,q,rg)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),h),hoare_new_addrs(h2,hoare_new_addrs(h3,as,h2),h)))) ).

% POST_G
tff(fact_4_POST__F,axiom,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(a,assn,r,rf)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),h2),hoare_new_addrs(h3,as,h2)))) ).

% POST_F
tff(fact_5_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( rep_assn(X) = rep_assn(Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_6_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_7_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
    <=> ( ( A3 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_8_one__assn__raw_Ocases,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ~ ! [H2: heap_ext(product_unit),As2: set(nat)] : X != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) ).

% one_assn_raw.cases
tff(fact_9_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,X: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F: fun(A,B),A5: A,B4: A] : X != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)) ).

% pairself.cases
tff(fact_10_snga__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R: array(A),X3: list(A),H2: heap_ext(product_unit),As2: set(nat)] : X != aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),R),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))) ) ).

% snga_assn_raw.cases
tff(fact_11_sngr__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R: ref(A),X3: A,H2: heap_ext(product_unit),As2: set(nat)] : X != aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),R),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))) ) ).

% sngr_assn_raw.cases
tff(fact_12_times__assn__raw_Ocases,axiom,
    ! [X: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))] :
      ~ ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H2: heap_ext(product_unit),As2: set(nat)] : X != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),P),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))) ).

% times_assn_raw.cases
tff(fact_13_pure__assn__raw_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(bool,product_prod(A,set(B)))] :
      ~ ! [B4: bool,H2: A,As2: set(B)] : X != aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),B4),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2)) ).

% pure_assn_raw.cases
tff(fact_14_prod__induct7,axiom,
    ! [G: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))))] :
      ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F: F2,G2: G] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F2,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G)))),C2),aa(product_prod(E,product_prod(F2,G)),product_prod(D,product_prod(E,product_prod(F2,G))),aa(D,fun(product_prod(E,product_prod(F2,G)),product_prod(D,product_prod(E,product_prod(F2,G)))),product_Pair(D,product_prod(E,product_prod(F2,G))),D2),aa(product_prod(F2,G),product_prod(E,product_prod(F2,G)),aa(E,fun(product_prod(F2,G),product_prod(E,product_prod(F2,G))),product_Pair(E,product_prod(F2,G)),E2),aa(G,product_prod(F2,G),aa(F2,fun(G,product_prod(F2,G)),product_Pair(F2,G),F),G2))))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),bool,P2,X)) ) ).

% prod_induct7
tff(fact_15_prod__induct6,axiom,
    ! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
      ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F: F2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B4),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E2),F)))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P2,X)) ) ).

% prod_induct6
tff(fact_16_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P2,X)) ) ).

% prod_induct5
tff(fact_17_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool),X: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A5: A,B4: B,C2: C,D2: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P2,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2)))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P2,X)) ) ).

% prod_induct4
tff(fact_18_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,C)),bool),X: product_prod(A,product_prod(B,C))] :
      ( ! [A5: A,B4: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P2,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)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2))))
     => pp(aa(product_prod(A,product_prod(B,C)),bool,P2,X)) ) ).

% prod_induct3
tff(fact_19_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F: F2,G2: G] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F2,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G)))),C2),aa(product_prod(E,product_prod(F2,G)),product_prod(D,product_prod(E,product_prod(F2,G))),aa(D,fun(product_prod(E,product_prod(F2,G)),product_prod(D,product_prod(E,product_prod(F2,G)))),product_Pair(D,product_prod(E,product_prod(F2,G))),D2),aa(product_prod(F2,G),product_prod(E,product_prod(F2,G)),aa(E,fun(product_prod(F2,G),product_prod(E,product_prod(F2,G))),product_Pair(E,product_prod(F2,G)),E2),aa(G,product_prod(F2,G),aa(F2,fun(G,product_prod(F2,G)),product_Pair(F2,G),F),G2)))))) ).

% prod_cases7
tff(fact_20_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F: F2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B4),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E2),F))))) ).

% prod_cases6
tff(fact_21_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).

% prod_cases5
tff(fact_22_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2))) ).

% prod_cases4
tff(fact_23_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A5: A,B4: B,C2: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)) ).

% prod_cases3
tff(fact_24_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
     => ~ ( ( A3 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_25_prod__cases,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool),P3: product_prod(A,B)] :
      ( ! [A5: A,B4: B] : pp(aa(product_prod(A,B),bool,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
     => pp(aa(product_prod(A,B),bool,P2,P3)) ) ).

% prod_cases
tff(fact_26_surj__pair,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B)] :
    ? [X3: A,Y3: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) ).

% surj_pair
tff(fact_27_bex2I,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,S: set(product_prod(A,B)),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
     => ( ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,A3),B2)) )
       => ? [A5: A,B4: B] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),S))
            & pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) ) ) ) ).

% bex2I
tff(fact_28_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A5: A,B4: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) ).

% old.prod.exhaust
tff(fact_29_T1,axiom,
    hoare_hoare_triple(a,p,f,r) ).

% T1
tff(fact_30_old_Oprod_Orec,axiom,
    ! [A: $tType,T: $tType,B: $tType,F1: fun(A,fun(B,T)),A3: A,B2: B] : product_rec_prod(A,B,T,F1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) = aa(B,T,aa(A,fun(B,T),F1,A3),B2) ).

% old.prod.rec
tff(fact_31_T2,axiom,
    ! [X: a] : hoare_hoare_triple(b,aa(a,assn,r,X),g(X),q) ).

% T2
tff(fact_32_EX__G,axiom,
    aa(heap_ext(product_unit),option(product_prod(b,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(b,g(rf)),h2) = aa(product_prod(b,product_prod(heap_ext(product_unit),nat)),option(product_prod(b,product_prod(heap_ext(product_unit),nat))),some(product_prod(b,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(b,product_prod(heap_ext(product_unit),nat)),aa(b,fun(product_prod(heap_ext(product_unit),nat),product_prod(b,product_prod(heap_ext(product_unit),nat))),product_Pair(b,product_prod(heap_ext(product_unit),nat)),rg),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),h),t))) ).

% EX_G
tff(fact_33_LIM__G,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,h2)),lim(product_unit,h))) ).

% LIM_G
tff(fact_34_uncurry__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,uncurry(B,C,A,F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F3,A3),B2) ).

% uncurry_apply
tff(fact_35_LIM__F,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,h3)),lim(product_unit,h2))) ).

% LIM_F
tff(fact_36_internal__case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc5280177257484947105e_prod(B,C,A),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),C3,A3),B2) ).

% internal_case_prod_conv
tff(fact_37_EX__F,axiom,
    aa(heap_ext(product_unit),option(product_prod(a,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(a,f),h3) = aa(product_prod(a,product_prod(heap_ext(product_unit),nat)),option(product_prod(a,product_prod(heap_ext(product_unit),nat))),some(product_prod(a,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(a,product_prod(heap_ext(product_unit),nat)),aa(a,fun(product_prod(heap_ext(product_unit),nat),product_prod(a,product_prod(heap_ext(product_unit),nat))),product_Pair(a,product_prod(heap_ext(product_unit),nat)),rf),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),h2),t2))) ).

% EX_F
tff(fact_38_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(fun(A,bool),set(A),collect(A),P2)))
    <=> pp(aa(A,bool,P2,A3)) ) ).

% mem_Collect_eq
tff(fact_39_Collect__mem__eq,axiom,
    ! [A: $tType,A6: set(A)] : aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)) = A6 ).

% Collect_mem_eq
tff(fact_40_Collect__cong,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
        <=> pp(aa(A,bool,Q2,X3)) )
     => ( aa(fun(A,bool),set(A),collect(A),P2) = aa(fun(A,bool),set(A),collect(A),Q2) ) ) ).

% Collect_cong
tff(fact_41_ext,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F3,X3) = aa(A,B,G3,X3)
     => ( F3 = G3 ) ) ).

% ext
tff(fact_42_bijective__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( bijective(A,B,R2)
    <=> ( ! [X4: A,Y4: B,Z2: B] :
            ( ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4)),R2))
              & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z2)),R2)) )
           => ( Y4 = Z2 ) )
        & ! [X4: A,Y4: A,Z2: B] :
            ( ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z2)),R2))
              & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y4),Z2)),R2)) )
           => ( X4 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_43_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R3: A,S2: B,R2: set(product_prod(A,B)),S3: B] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R3),S2)),R2))
     => ( ( S3 = S2 )
       => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R3),S3)),R2)) ) ) ).

% ssubst_Pair_rhs
tff(fact_44_curry__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(product_prod(B,C),A),A3: B,B2: C] : aa(C,A,aa(B,fun(C,A),aa(fun(product_prod(B,C),A),fun(B,fun(C,A)),product_curry(B,C,A),F3),A3),B2) = aa(product_prod(B,C),A,F3,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) ).

% curry_conv
tff(fact_45_curryI,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),product_curry(A,B,bool),F3),A3),B2)) ) ).

% curryI
tff(fact_46_curry__uncurry__id,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),uncurry(A,B,C,F3)) = F3 ).

% curry_uncurry_id
tff(fact_47_uncurry__curry__id,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : uncurry(A,B,C,aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),F3)) = F3 ).

% uncurry_curry_id
tff(fact_48_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,X: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),M)),X))
         => ~ ! [M2: A] :
                ( ( X = aa(A,option(A),some(A),M2) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),M2)) ) ) ) ).

% le_some_optE
tff(fact_49_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => ( ( C3 = D3 )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),D3)) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_50_hoare__triple__preI,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H2))
         => hoare_hoare_triple(A,P2,C3,Q2) )
     => hoare_hoare_triple(A,P2,C3,Q2) ) ).

% hoare_triple_preI
tff(fact_51_curryD,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),product_curry(A,B,bool),F3),A3),B2))
     => pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryD
tff(fact_52_curryE,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),product_curry(A,B,bool),F3),A3),B2))
     => pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryE
tff(fact_53_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% less_eq_option_Some
tff(fact_54_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_55_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).

% dual_order.refl
tff(fact_56_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).

% order_refl
tff(fact_57__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rg_Ah_H_H_At_H_H_O_A_092_060lbrakk_062execute_A_Ig_Arf_J_Ah_H_A_061_ASome_A_Irg_M_Ah_H_H_M_At_H_H_J_059_A_Ih_H_H_M_Anew__addrs_Ah_H_A_Inew__addrs_Ah_Aas_Ah_H_J_Ah_H_H_J_A_092_060Turnstile_062_AQ_Arg_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_H_A_092_060and_062_Aa_A_092_060notin_062_Anew__addrs_Ah_Aas_Ah_H_125_Ah_H_Ah_H_H_059_Alim_Ah_H_A_092_060le_062_Alim_Ah_H_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Rg: b,H3: heap_ext(product_unit)] :
        ( ? [T2: nat] : aa(heap_ext(product_unit),option(product_prod(b,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(b,g(rf)),h2) = aa(product_prod(b,product_prod(heap_ext(product_unit),nat)),option(product_prod(b,product_prod(heap_ext(product_unit),nat))),some(product_prod(b,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(b,product_prod(heap_ext(product_unit),nat)),aa(b,fun(product_prod(heap_ext(product_unit),nat),product_prod(b,product_prod(heap_ext(product_unit),nat))),product_Pair(b,product_prod(heap_ext(product_unit),nat)),Rg),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H3),T2)))
       => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(b,assn,q,Rg)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),hoare_new_addrs(h2,hoare_new_addrs(h3,as,h2),H3))))
         => ( relH(aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aa(nat,bool)),h2,H3)
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,h2)),lim(product_unit,H3))) ) ) ) ).

% \<open>\<And>thesis. (\<And>rg h'' t''. \<lbrakk>execute (g rf) h' = Some (rg, h'', t''); (h'', new_addrs h' (new_addrs h as h') h'') \<Turnstile> Q rg; relH {a. a < lim h' \<and> a \<notin> new_addrs h as h'} h' h''; lim h' \<le> lim h''\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_58__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rf_Ah_H_At_H_O_A_092_060lbrakk_062execute_Af_Ah_A_061_ASome_A_Irf_M_Ah_H_M_At_H_J_059_A_Ih_H_M_Anew__addrs_Ah_Aas_Ah_H_J_A_092_060Turnstile_062_AR_Arf_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_Ah_Ah_H_059_Alim_Ah_A_092_060le_062_Alim_Ah_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Rf: a,H4: heap_ext(product_unit)] :
        ( ? [T3: nat] : aa(heap_ext(product_unit),option(product_prod(a,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(a,f),h3) = aa(product_prod(a,product_prod(heap_ext(product_unit),nat)),option(product_prod(a,product_prod(heap_ext(product_unit),nat))),some(product_prod(a,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(a,product_prod(heap_ext(product_unit),nat)),aa(a,fun(product_prod(heap_ext(product_unit),nat),product_prod(a,product_prod(heap_ext(product_unit),nat))),product_Pair(a,product_prod(heap_ext(product_unit),nat)),Rf),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),T3)))
       => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(a,assn,r,Rf)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),hoare_new_addrs(h3,as,H4))))
         => ( relH(aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ab(nat,bool)),h3,H4)
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,h3)),lim(product_unit,H4))) ) ) ) ).

% \<open>\<And>thesis. (\<And>rf h' t'. \<lbrakk>execute f h = Some (rf, h', t'); (h', new_addrs h as h') \<Turnstile> R rf; relH {a. a < lim h \<and> a \<notin> as} h h'; lim h \<le> lim h'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_59_the__default_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: A,X: A] : the_default(A,Uu,aa(A,option(A),some(A),X)) = X ).

% the_default.simps(1)
tff(fact_60_hoare__triple__effect,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn),H: heap_ext(product_unit),As: set(nat)] :
      ( hoare_hoare_triple(A,P2,C3,Q2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
       => ? [H4: heap_ext(product_unit),R: A,T4: nat] :
            ( heap_Time_effect(A,C3,H,H4,R,T4)
            & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q2,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),hoare_new_addrs(H,As,H4)))) ) ) ) ).

% hoare_triple_effect
tff(fact_61_hoare__triple__success,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn),H: heap_ext(product_unit),As: set(nat)] :
      ( hoare_hoare_triple(A,P2,C3,Q2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
       => heap_Time_success(A,C3,H) ) ) ).

% hoare_triple_success
tff(fact_62_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R3: set(product_prod(A,A)),As3: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R3,As3)
        <=> ! [I: A,J: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As3,I)),aa(A,B,As3,J))) ) ) ) ).

% relChain_def
tff(fact_63_Heap__eqI,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),G3: heap_Time_Heap(A)] :
      ( ! [H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H2) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,G3),H2)
     => ( F3 = G3 ) ) ).

% Heap_eqI
tff(fact_64_RH__G,axiom,
    relH(aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aa(nat,bool)),h2,h) ).

% RH_G
tff(fact_65_RH__F,axiom,
    relH(aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ab(nat,bool)),h3,h2) ).

% RH_F
tff(fact_66_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),aa(A,option(A),some(A),X)),aa(A,option(A),some(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% less_option_Some
tff(fact_67_relH__subset,axiom,
    ! [Bs: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),As: set(nat)] :
      ( relH(Bs,H,H5)
     => ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As),Bs))
       => relH(As,H,H5) ) ) ).

% relH_subset
tff(fact_68_effect__deterministic_I3_J,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),A3: A,N: nat,H6: heap_ext(product_unit),B2: A,N2: nat] :
      ( heap_Time_effect(A,F3,H,H5,A3,N)
     => ( heap_Time_effect(A,F3,H,H6,B2,N2)
       => ( N = N2 ) ) ) ).

% effect_deterministic(3)
tff(fact_69_effect__deterministic_I2_J,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),A3: A,N: nat,H6: heap_ext(product_unit),B2: A,N2: nat] :
      ( heap_Time_effect(A,F3,H,H5,A3,N)
     => ( heap_Time_effect(A,F3,H,H6,B2,N2)
       => ( H5 = H6 ) ) ) ).

% effect_deterministic(2)
tff(fact_70_effect__deterministic_I1_J,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),A3: A,N: nat,H6: heap_ext(product_unit),B2: A,N2: nat] :
      ( heap_Time_effect(A,F3,H,H5,A3,N)
     => ( heap_Time_effect(A,F3,H,H6,B2,N2)
       => ( A3 = B2 ) ) ) ).

% effect_deterministic(1)
tff(fact_71_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X)) ) ).

% lt_ex
tff(fact_72_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_1)) ) ).

% gt_ex
tff(fact_73_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),Y)) ) ) ) ).

% dense
tff(fact_74_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_75_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order.asym
tff(fact_76_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% ord_eq_less_trans
tff(fact_77_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( B2 = C3 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% ord_less_eq_trans
tff(fact_78_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X3))
                 => pp(aa(A,bool,P2,Y5)) )
             => pp(aa(A,bool,P2,X3)) )
         => pp(aa(A,bool,P2,A3)) ) ) ).

% less_induct
tff(fact_79_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_80_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_cases
tff(fact_81_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% dual_order.asym
tff(fact_82_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).

% dual_order.irrefl
tff(fact_83_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool)] :
          ( ? [X_12: A] : pp(aa(A,bool,P2,X_12))
        <=> ? [N3: A] :
              ( pp(aa(A,bool,P2,N3))
              & ! [M3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M3),N3))
                 => ~ pp(aa(A,bool,P2,M3)) ) ) ) ) ).

% exists_least_iff
tff(fact_84_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,bool)),A3: A,B2: A] :
          ( ! [A5: A,B4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
         => ( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),A5))
           => ( ! [A5: A,B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),P2,B4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,A3),B2)) ) ) ) ) ).

% linorder_less_wlog
tff(fact_85_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% order.strict_trans
tff(fact_86_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_87_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% dual_order.strict_trans
tff(fact_88_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( A3 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_89_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( A3 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_90_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE
tff(fact_91_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_asym
tff(fact_92_effect__ifE,axiom,
    ! [A: $tType,C3: bool,T5: heap_Time_Heap(A),E3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,if(heap_Time_Heap(A),C3,T5,E3),H,H5,R3,N)
     => ( ( pp(C3)
         => ~ heap_Time_effect(A,T5,H,H5,R3,N) )
       => ~ ( ~ pp(C3)
           => ~ heap_Time_effect(A,E3,H,H5,R3,N) ) ) ) ).

% effect_ifE
tff(fact_93_effect__ifI,axiom,
    ! [A: $tType,C3: bool,T5: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat,E3: heap_Time_Heap(A)] :
      ( ( pp(C3)
       => heap_Time_effect(A,T5,H,H5,R3,N) )
     => ( ( ~ pp(C3)
         => heap_Time_effect(A,E3,H,H5,R3,N) )
       => heap_Time_effect(A,if(heap_Time_Heap(A),C3,T5,E3),H,H5,R3,N) ) ) ).

% effect_ifI
tff(fact_94_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neq_iff
tff(fact_95_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order_less_asym'
tff(fact_96_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z4)) ) ) ) ).

% order_less_trans
tff(fact_97_effect__LetI,axiom,
    ! [B: $tType,A: $tType,X: A,T5: A,F3: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: B,N: nat] :
      ( ( X = T5 )
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),F3,X),H,H5,R3,N)
       => heap_Time_effect(B,aa(A,heap_Time_Heap(B),F3,T5),H,H5,R3,N) ) ) ).

% effect_LetI
tff(fact_98_success__ifI,axiom,
    ! [A: $tType,C3: bool,T5: heap_Time_Heap(A),H: heap_ext(product_unit),E3: heap_Time_Heap(A)] :
      ( ( pp(C3)
       => heap_Time_success(A,T5,H) )
     => ( ( ~ pp(C3)
         => heap_Time_success(A,E3,H) )
       => heap_Time_success(A,if(heap_Time_Heap(A),C3,T5,E3),H) ) ) ).

% success_ifI
tff(fact_99_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A3 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_100_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F3: fun(A,B),C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,A3)),C3)) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_101_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).

% order_less_irrefl
tff(fact_102_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_less_subst1
tff(fact_103_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_less_subst2
tff(fact_104_success__LetI,axiom,
    ! [A: $tType,B: $tType,X: A,T5: A,F3: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
      ( ( X = T5 )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),F3,X),H)
       => heap_Time_success(B,aa(A,heap_Time_Heap(B),F3,T5),H) ) ) ).

% success_LetI
tff(fact_105_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_not_sym
tff(fact_106_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => pp(P2) ) ) ) ).

% order_less_imp_triv
tff(fact_107_effect__success,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,C3,H,H5,R3,N)
     => heap_Time_success(A,C3,H) ) ).

% effect_success
tff(fact_108_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          | ( X = Y )
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_less_linear
tff(fact_109_success__effectE,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,C3,H)
     => ~ ! [R: A,H4: heap_ext(product_unit),N4: nat] : ~ heap_Time_effect(A,C3,H,H4,R,N4) ) ).

% success_effectE
tff(fact_110_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_111_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_112_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_imp_not_less
tff(fact_113_relH__trans,axiom,
    ! [As: set(nat),H1: heap_ext(product_unit),H22: heap_ext(product_unit),H32: heap_ext(product_unit)] :
      ( relH(As,H1,H22)
     => ( relH(As,H22,H32)
       => relH(As,H1,H32) ) ) ).

% relH_trans
tff(fact_114_relH__sym,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
      ( relH(As,H,H5)
     => relH(As,H5,H) ) ).

% relH_sym
tff(fact_115_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% leD
tff(fact_116_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% leI
tff(fact_117_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% nless_le
tff(fact_118_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_119_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_120_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z4: A,Y: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4)) ) ) ).

% dense_ge
tff(fact_121_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z4: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4)) ) ) ).

% dense_le
tff(fact_122_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% less_le_not_le
tff(fact_123_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% not_le_imp_less
tff(fact_124_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_125_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_126_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% order.strict_trans1
tff(fact_127_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% order.strict_trans2
tff(fact_128_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% order.strict_iff_not
tff(fact_129_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z4: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),X))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),W)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4)) ) ) ) ).

% dense_ge_bounded
tff(fact_130_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z4)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4)) ) ) ) ).

% dense_le_bounded
tff(fact_131_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
            | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_132_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_133_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_134_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_135_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_136_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% order.strict_implies_order
tff(fact_137_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% dual_order.strict_implies_order
tff(fact_138_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_139_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_140_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_not_le
tff(fact_141_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_not_less
tff(fact_142_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_less_imp_le
tff(fact_143_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( A3 != B2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_144_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_145_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z4)) ) ) ) ).

% order_le_less_trans
tff(fact_146_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z4)) ) ) ) ).

% order_less_le_trans
tff(fact_147_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_148_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_149_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_150_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_151_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_le_less_linear
tff(fact_152_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_153_curry__K,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: C,X5: A,Xa: B] : aa(B,C,aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),aTP_Lamp_ac(C,fun(product_prod(A,B),C),C3)),X5),Xa) = C3 ).

% curry_K
tff(fact_154_curry__def,axiom,
    ! [C: $tType,A: $tType,B: $tType,X5: fun(product_prod(A,B),C),Xa: A,Xb: B] : aa(B,C,aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),X5),Xa),Xb) = aa(product_prod(A,B),C,X5,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)) ).

% curry_def
tff(fact_155_exists__leI,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ( ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),N))
           => ~ pp(aa(nat,bool,P2,N5)) )
       => pp(aa(nat,bool,P2,N)) )
     => ? [N6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N))
          & pp(aa(nat,bool,P2,N6)) ) ) ).

% exists_leI
tff(fact_156_mod__relH,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),P2: assn] :
      ( relH(As,H,H5)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
      <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))) ) ) ).

% mod_relH
tff(fact_157_hoare__triple__def,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( hoare_hoare_triple(A,P2,C3,Q2)
    <=> ! [H7: heap_ext(product_unit),As4: set(nat)] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As4)))
         => ? [R4: A,H8: heap_ext(product_unit)] :
              ( ? [T6: nat] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H7) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R4),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H8),T6)))
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q2,R4)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),hoare_new_addrs(H7,As4,H8))))
              & relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ad(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H7),As4)),H7,H8)
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H7)),lim(product_unit,H8))) ) ) ) ).

% hoare_triple_def
tff(fact_158_hoare__tripleI,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( ! [H2: heap_ext(product_unit),As2: set(nat)] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
         => ? [R5: A,H9: heap_ext(product_unit),T7: nat] :
              ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R5),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H9),T7))) )
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q2,R5)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),hoare_new_addrs(H2,As2,H9))))
              & relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ad(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H2),As2)),H2,H9)
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H2)),lim(product_unit,H9))) ) )
     => hoare_hoare_triple(A,P2,C3,Q2) ) ).

% hoare_tripleI
tff(fact_159_hoare__tripleE,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn),H: heap_ext(product_unit),As: set(nat)] :
      ( hoare_hoare_triple(A,P2,C3,Q2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
       => ~ ! [R: A,H4: heap_ext(product_unit)] :
              ( ? [T4: nat] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),T4)))
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Q2,R)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),hoare_new_addrs(H,As,H4))))
               => ( relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ad(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H),As)),H,H4)
                 => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H)),lim(product_unit,H4))) ) ) ) ) ) ).

% hoare_tripleE
tff(fact_160_successE,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F3,H)
     => ~ ! [R: A,H4: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),H4)) ) ).

% successE
tff(fact_161_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ( B2 != A3 ) ) ) ) ).

% nle_le
tff(fact_162_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4)) )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z4)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z4))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),Y)) )
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),Y))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
               => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4))
                   => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X)) )
                 => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X))
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_163_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% order_class.order_eq_iff
tff(fact_164_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% ord_eq_le_trans
tff(fact_165_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( B2 = C3 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% ord_le_eq_trans
tff(fact_166_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_167_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% order.trans
tff(fact_168_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z4)) ) ) ) ).

% order_trans
tff(fact_169_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,bool)),A3: A,B2: A] :
          ( ! [A5: A,B4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
         => ( ! [A5: A,B4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),P2,B4),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,A3),B2)) ) ) ) ).

% linorder_wlog
tff(fact_170_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% dual_order.eq_iff
tff(fact_171_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( A3 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_172_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% dual_order.trans
tff(fact_173_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => ( A3 = B2 ) ) ) ) ).

% antisym
tff(fact_174_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))) ) ) ).

% le_funD
tff(fact_175_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))) ) ) ).

% le_funE
tff(fact_176_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)))
         => pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3)) ) ) ).

% le_funI
tff(fact_177_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
        <=> ! [X4: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4))) ) ) ).

% le_fun_def
tff(fact_178_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_179_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_subst1
tff(fact_180_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_subst2
tff(fact_181_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_eq_refl
tff(fact_182_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_linear
tff(fact_183_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A3 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_184_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F3: fun(A,B),C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),C3)) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_185_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_le_cases
tff(fact_186_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_187_effect__def,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,C3,H,H5,R3,N)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H5),N))) ) ) ).

% effect_def
tff(fact_188_effectI,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),H: heap_ext(product_unit),R3: A,H5: heap_ext(product_unit),N: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H5),N))) )
     => heap_Time_effect(A,C3,H,H5,R3,N) ) ).

% effectI
tff(fact_189_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_190_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_imp_le_nat
tff(fact_191_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_192_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_or_eq_imp_le
tff(fact_193_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ( M != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% le_neq_implies_less
tff(fact_194_less__mono__imp__le__mono,axiom,
    ! [F3: fun(nat,nat),I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,I3)),aa(nat,nat,F3,J3))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F3,I2)),aa(nat,nat,F3,J2))) ) ) ).

% less_mono_imp_le_mono
tff(fact_195_nat__descend__induct,axiom,
    ! [N: nat,P2: fun(nat,bool),M: nat] :
      ( ! [K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
         => pp(aa(nat,bool,P2,K)) )
     => ( ! [K: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( ! [I4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),I4))
                 => pp(aa(nat,bool,P2,I4)) )
             => pp(aa(nat,bool,P2,K)) ) )
       => pp(aa(nat,bool,P2,M)) ) ) ).

% nat_descend_induct
tff(fact_196_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P2,K2))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => ? [X3: A] :
            ( pp(aa(A,bool,P2,X3))
            & ! [Y5: A] :
                ( pp(aa(A,bool,P2,Y5))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y5)),aa(A,nat,F3,X3))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_197_success__bind__executeI,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H5: heap_ext(product_unit),N: nat,G3: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H5),N))) )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G3,X),H5)
       => heap_Time_success(B,heap_Time_bind(A,B,F3,G3),H) ) ) ).

% success_bind_executeI
tff(fact_198_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T5),X5)) ) ) ).

% minf(8)
tff(fact_199_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),T5)) ) ) ).

% minf(6)
tff(fact_200_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T5),X5)) ) ) ).

% pinf(8)
tff(fact_201_Heap__Time__Monad_Obind__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: heap_Time_Heap(A),G3: fun(A,heap_Time_Heap(C)),K2: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,heap_Time_bind(A,C,F3,G3),K2) = heap_Time_bind(A,B,F3,aa(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aTP_Lamp_ae(fun(A,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),G3),K2)) ).

% Heap_Time_Monad.bind_bind
tff(fact_202_subset__Collect__conv,axiom,
    ! [A: $tType,S: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(fun(A,bool),set(A),collect(A),P2)))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S))
         => pp(aa(A,bool,P2,X4)) ) ) ).

% subset_Collect_conv
tff(fact_203_Collect__restrict,axiom,
    ! [A: $tType,X6: set(A),P2: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),X6),P2))),X6)) ).

% Collect_restrict
tff(fact_204_prop__restrict,axiom,
    ! [A: $tType,X: A,Z5: set(A),X6: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,X),Z5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z5),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),X6),P2))))
       => pp(aa(A,bool,P2,X)) ) ) ).

% prop_restrict
tff(fact_205_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F3),G3))
        <=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
            & ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G3),F3)) ) ) ) ).

% less_fun_def
tff(fact_206_distrib__if__bind,axiom,
    ! [A: $tType,B: $tType,B2: bool,C3: heap_Time_Heap(B),D3: heap_Time_Heap(B),F3: fun(B,heap_Time_Heap(A))] :
      ( ( pp(B2)
       => ( heap_Time_bind(B,A,if(heap_Time_Heap(B),B2,C3,D3),F3) = heap_Time_bind(B,A,C3,F3) ) )
      & ( ~ pp(B2)
       => ( heap_Time_bind(B,A,if(heap_Time_Heap(B),B2,C3,D3),F3) = heap_Time_bind(B,A,D3,F3) ) ) ) ).

% distrib_if_bind
tff(fact_207_success__bind__effectI,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),X: A,N: nat,G3: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_effect(A,F3,H,H5,X,N)
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G3,X),H5)
       => heap_Time_success(B,heap_Time_bind(A,B,F3,G3),H) ) ) ).

% success_bind_effectI
tff(fact_208_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y5: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y5)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P2,Y5)) )
             => pp(aa(A,bool,P2,X3)) )
         => pp(aa(A,bool,P2,A3)) ) ) ).

% measure_induct_rule
tff(fact_209_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y5: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y5)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P2,Y5)) )
             => pp(aa(A,bool,P2,X3)) )
         => pp(aa(A,bool,P2,A3)) ) ) ).

% measure_induct
tff(fact_210_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool),P4: fun(A,bool),Q2: fun(A,bool),Q3: fun(A,bool)] :
          ( ? [Z6: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z6),X3))
             => ( pp(aa(A,bool,P2,X3))
              <=> pp(aa(A,bool,P4,X3)) ) )
         => ( ? [Z6: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z6),X3))
               => ( pp(aa(A,bool,Q2,X3))
                <=> pp(aa(A,bool,Q3,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
               => ( ( pp(aa(A,bool,P2,X5))
                    & pp(aa(A,bool,Q2,X5)) )
                <=> ( pp(aa(A,bool,P4,X5))
                    & pp(aa(A,bool,Q3,X5)) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_211_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool),P4: fun(A,bool),Q2: fun(A,bool),Q3: fun(A,bool)] :
          ( ? [Z6: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z6),X3))
             => ( pp(aa(A,bool,P2,X3))
              <=> pp(aa(A,bool,P4,X3)) ) )
         => ( ? [Z6: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z6),X3))
               => ( pp(aa(A,bool,Q2,X3))
                <=> pp(aa(A,bool,Q3,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
               => ( ( pp(aa(A,bool,P2,X5))
                    | pp(aa(A,bool,Q2,X5)) )
                <=> ( pp(aa(A,bool,P4,X5))
                    | pp(aa(A,bool,Q3,X5)) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_212_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ( X5 != T5 ) ) ) ).

% pinf(3)
tff(fact_213_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ( X5 != T5 ) ) ) ).

% pinf(4)
tff(fact_214_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),T5)) ) ) ).

% pinf(5)
tff(fact_215_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T5),X5)) ) ) ).

% pinf(7)
tff(fact_216_pinf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X5: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),Z3),X5))
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_217_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool),P4: fun(A,bool),Q2: fun(A,bool),Q3: fun(A,bool)] :
          ( ? [Z6: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z6))
             => ( pp(aa(A,bool,P2,X3))
              <=> pp(aa(A,bool,P4,X3)) ) )
         => ( ? [Z6: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z6))
               => ( pp(aa(A,bool,Q2,X3))
                <=> pp(aa(A,bool,Q3,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
               => ( ( pp(aa(A,bool,P2,X5))
                    & pp(aa(A,bool,Q2,X5)) )
                <=> ( pp(aa(A,bool,P4,X5))
                    & pp(aa(A,bool,Q3,X5)) ) ) ) ) ) ) ).

% minf(1)
tff(fact_218_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool),P4: fun(A,bool),Q2: fun(A,bool),Q3: fun(A,bool)] :
          ( ? [Z6: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z6))
             => ( pp(aa(A,bool,P2,X3))
              <=> pp(aa(A,bool,P4,X3)) ) )
         => ( ? [Z6: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z6))
               => ( pp(aa(A,bool,Q2,X3))
                <=> pp(aa(A,bool,Q3,X3)) ) )
           => ? [Z3: A] :
              ! [X5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
               => ( ( pp(aa(A,bool,P2,X5))
                    | pp(aa(A,bool,Q2,X5)) )
                <=> ( pp(aa(A,bool,P4,X5))
                    | pp(aa(A,bool,Q3,X5)) ) ) ) ) ) ) ).

% minf(2)
tff(fact_219_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ( X5 != T5 ) ) ) ).

% minf(3)
tff(fact_220_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ( X5 != T5 ) ) ) ).

% minf(4)
tff(fact_221_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),T5)) ) ) ).

% minf(5)
tff(fact_222_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T5),X5)) ) ) ).

% minf(7)
tff(fact_223_minf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F4: D] :
        ? [Z3: C] :
        ! [X5: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),X5),Z3))
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_224_infinite__descent__measure,axiom,
    ! [A: $tType,P2: fun(A,bool),V: fun(A,nat),X: A] :
      ( ! [X3: A] :
          ( ~ pp(aa(A,bool,P2,X3))
         => ? [Y5: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y5)),aa(A,nat,V,X3)))
              & ~ pp(aa(A,bool,P2,Y5)) ) )
     => pp(aa(A,bool,P2,X)) ) ).

% infinite_descent_measure
tff(fact_225_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_226_infinite__descent,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( ! [N4: nat] :
          ( ~ pp(aa(nat,bool,P2,N4))
         => ? [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N4))
              & ~ pp(aa(nat,bool,P2,M4)) ) )
     => pp(aa(nat,bool,P2,N)) ) ).

% infinite_descent
tff(fact_227_nat__less__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( ! [N4: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N4))
             => pp(aa(nat,bool,P2,M4)) )
         => pp(aa(nat,bool,P2,N4)) )
     => pp(aa(nat,bool,P2,N)) ) ).

% nat_less_induct
tff(fact_228_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_irrefl_nat
tff(fact_229_less__not__refl3,axiom,
    ! [S2: nat,T5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T5))
     => ( S2 != T5 ) ) ).

% less_not_refl3
tff(fact_230_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_231_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_not_refl
tff(fact_232_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).

% nat_neq_iff
tff(fact_233_ex__has__least__nat,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P2,K2))
     => ? [X3: A] :
          ( pp(aa(A,bool,P2,X3))
          & ! [Y5: A] :
              ( pp(aa(A,bool,P2,Y5))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,X3)),aa(A,nat,M,Y5))) ) ) ) ).

% ex_has_least_nat
tff(fact_234_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun(nat,bool),K2: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => ? [X3: nat] :
            ( pp(aa(nat,bool,P2,X3))
            & ! [Y5: nat] :
                ( pp(aa(nat,bool,P2,Y5))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y5),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_235_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% nat_le_linear
tff(fact_236_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_237_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% eq_imp_le
tff(fact_238_le__trans,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K2)) ) ) ).

% le_trans
tff(fact_239_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

% le_refl
tff(fact_240_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T5: A] :
        ? [Z3: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X5))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),T5)) ) ) ).

% pinf(6)
tff(fact_241_execute__bind_I1_J,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H5: heap_ext(product_unit),N: nat,G3: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H5),N))) )
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F3,G3)),H) = heap_Time_timeFrame(B,N,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G3,X)),H5)) ) ) ).

% execute_bind(1)
tff(fact_242_subset__antisym,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
       => ( A6 = B5 ) ) ) ).

% subset_antisym
tff(fact_243_psubsetI,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( ( A6 != B5 )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5)) ) ) ).

% psubsetI
tff(fact_244_subsetI,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(set(A),bool,member(A,X3),B5)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ).

% subsetI
tff(fact_245_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B2: A,P2: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,P2,A3))
           => ( ~ pp(aa(A,bool,P2,B2))
             => ? [C2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
                  & ! [X5: A] :
                      ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X5))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),C2)) )
                     => pp(aa(A,bool,P2,X5)) )
                  & ! [D4: A] :
                      ( ! [X3: A] :
                          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),D4)) )
                         => pp(aa(A,bool,P2,X3)) )
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D4),C2)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_246_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B3: B,A4: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B3),A4))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A4),B3)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_247_execute__bind__eq__SomeI,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H5: heap_ext(product_unit),N: nat,G3: fun(A,heap_Time_Heap(B)),Y: B,H6: heap_ext(product_unit),N2: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H5),N))) )
     => ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G3,X)),H5) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H6),N2))) )
       => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F3,G3)),H) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H6),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_248_new__addrs__def,axiom,
    ! [H: heap_ext(product_unit),As: set(nat),H5: heap_ext(product_unit)] : hoare_new_addrs(H,As,H5) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),aa(fun(nat,bool),set(nat),collect(nat),aa(heap_ext(product_unit),fun(nat,bool),aTP_Lamp_ag(heap_ext(product_unit),fun(heap_ext(product_unit),fun(nat,bool)),H),H5))) ).

% new_addrs_def
tff(fact_249_subset__Collect__iff,axiom,
    ! [A: $tType,B5: set(A),A6: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B5))
           => pp(aa(A,bool,P2,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_250_subset__CollectI,axiom,
    ! [A: $tType,B5: set(A),A6: set(A),Q2: fun(A,bool),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),B5))
           => ( pp(aa(A,bool,Q2,X3))
             => pp(aa(A,bool,P2,X3)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),B5),Q2))),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2)))) ) ) ).

% subset_CollectI
tff(fact_251_conj__subset__def,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool),Q2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ah(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(fun(A,bool),set(A),collect(A),P2)))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(fun(A,bool),set(A),collect(A),Q2))) ) ) ).

% conj_subset_def
tff(fact_252_UnCI,axiom,
    ! [A: $tType,C3: A,B5: set(A),A6: set(A)] :
      ( ( ~ pp(aa(set(A),bool,member(A,C3),B5))
       => pp(aa(set(A),bool,member(A,C3),A6)) )
     => pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))) ) ).

% UnCI
tff(fact_253_Un__iff,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))
    <=> ( pp(aa(set(A),bool,member(A,C3),A6))
        | pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% Un_iff
tff(fact_254_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_255_Un__subset__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)),C4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4)) ) ) ).

% Un_subset_iff
tff(fact_256_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_257_sup__Some,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: 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),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_Some
tff(fact_258_timeFrame__assoc,axiom,
    ! [A: $tType,N: nat,N2: nat,F3: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,N,heap_Time_timeFrame(A,N2,F3)) = heap_Time_timeFrame(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2),F3) ).

% timeFrame_assoc
tff(fact_259_relH__dist__union,axiom,
    ! [As: set(nat),As5: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
      ( relH(aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5),H,H5)
    <=> ( relH(As,H,H5)
        & relH(As5,H,H5) ) ) ).

% relH_dist_union
tff(fact_260_psubsetD,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C3: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,member(A,C3),A6))
       => pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% psubsetD
tff(fact_261_less__set__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B5))) ) ).

% less_set_def
tff(fact_262_psubset__trans,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B5),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),C4)) ) ) ).

% psubset_trans
tff(fact_263_UnE,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))
     => ( ~ pp(aa(set(A),bool,member(A,C3),A6))
       => pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% UnE
tff(fact_264_UnI1,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),A6))
     => pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))) ) ).

% UnI1
tff(fact_265_UnI2,axiom,
    ! [A: $tType,C3: A,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),B5))
     => pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))) ) ).

% UnI2
tff(fact_266_bex__Un,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),P2: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))
          & pp(aa(A,bool,P2,X4)) )
    <=> ( ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
            & pp(aa(A,bool,P2,X4)) )
        | ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B5))
            & pp(aa(A,bool,P2,X4)) ) ) ) ).

% bex_Un
tff(fact_267_ball__Un,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),P2: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))
         => pp(aa(A,bool,P2,X4)) )
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => pp(aa(A,bool,P2,X4)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B5))
           => pp(aa(A,bool,P2,X4)) ) ) ) ).

% ball_Un
tff(fact_268_Un__assoc,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: 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)),sup_sup(set(A)),A6),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) ).

% Un_assoc
tff(fact_269_Un__absorb,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),A6) = A6 ).

% Un_absorb
tff(fact_270_Un__commute,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A6) ).

% Un_commute
tff(fact_271_Un__left__absorb,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) ).

% Un_left_absorb
tff(fact_272_Un__left__commute,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),C4)) ).

% Un_left_commute
tff(fact_273_Collect__disj__eq,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ai(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,bool),set(A),collect(A),P2)),aa(fun(A,bool),set(A),collect(A),Q2)) ).

% Collect_disj_eq
tff(fact_274_Un__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aj(set(A),fun(set(A),fun(A,bool)),A6),B5)) ).

% Un_def
tff(fact_275_Un__mono,axiom,
    ! [A: $tType,A6: set(A),C4: set(A),B5: set(A),D5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),D5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),D5))) ) ) ).

% Un_mono
tff(fact_276_Un__least,axiom,
    ! [A: $tType,A6: set(A),C4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)),C4)) ) ) ).

% Un_least
tff(fact_277_Un__upper1,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))) ).

% Un_upper1
tff(fact_278_Un__upper2,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))) ).

% Un_upper2
tff(fact_279_Un__absorb1,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = B5 ) ) ).

% Un_absorb1
tff(fact_280_Un__absorb2,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = A6 ) ) ).

% Un_absorb2
tff(fact_281_subset__UnE,axiom,
    ! [A: $tType,C4: set(A),A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))
     => ~ ! [A7: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A7),A6))
           => ! [B6: set(A)] :
                ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),B5))
               => ( C4 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A7),B6) ) ) ) ) ).

% subset_UnE
tff(fact_282_subset__Un__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = B5 ) ) ).

% subset_Un_eq
tff(fact_283_add__lessD1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K2)) ) ).

% add_lessD1
tff(fact_284_add__less__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),L))) ) ) ).

% add_less_mono
tff(fact_285_not__add__less1,axiom,
    ! [I2: nat,J2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),I2)) ).

% not_add_less1
tff(fact_286_not__add__less2,axiom,
    ! [J2: nat,I2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I2)),I2)) ).

% not_add_less2
tff(fact_287_add__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ).

% add_less_mono1
tff(fact_288_trans__less__add1,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),M))) ) ).

% trans_less_add1
tff(fact_289_trans__less__add2,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J2))) ) ).

% trans_less_add2
tff(fact_290_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),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),K2),N) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% less_add_eq_less
tff(fact_291_add__leE,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)),N))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ) ).

% add_leE
tff(fact_292_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% le_add1
tff(fact_293_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ).

% le_add2
tff(fact_294_add__leD1,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% add_leD1
tff(fact_295_add__leD2,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ).

% add_leD2
tff(fact_296_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),L))
     => ? [N4: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N4) ) ).

% le_Suc_ex
tff(fact_297_add__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),L))) ) ) ).

% add_le_mono
tff(fact_298_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ).

% add_le_mono1
tff(fact_299_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),M))) ) ).

% trans_le_add1
tff(fact_300_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J2))) ) ).

% trans_le_add2
tff(fact_301_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_302_timeFrame_Osimps_I1_J,axiom,
    ! [A: $tType,N: nat,R3: A,H: heap_ext(product_unit),N2: nat] : heap_Time_timeFrame(A,N,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),N2)))) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)))) ).

% timeFrame.simps(1)
tff(fact_303_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,F3: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P2,K2))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => ? [Y5: A] :
                ( pp(aa(A,bool,P2,Y5))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y5)),aa(A,nat,F3,X3))) ) )
       => ? [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,K2)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_304_mono__nat__linear__lb,axiom,
    ! [F3: fun(nat,nat),M: nat,K2: nat] :
      ( ! [M5: nat,N4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,M5)),aa(nat,nat,F3,N4))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F3,M)),K2)),aa(nat,nat,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)))) ) ).

% mono_nat_linear_lb
tff(fact_305_effect__bindE,axiom,
    ! [B: $tType,A: $tType,F3: heap_Time_Heap(B),G3: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit),H6: heap_ext(product_unit),R6: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_bind(B,A,F3,G3),H,H6,R6,N)
     => ~ ! [H4: heap_ext(product_unit),R: B,N1: nat] :
            ( heap_Time_effect(B,F3,H,H4,R,N1)
           => ! [N22: nat] :
                ( heap_Time_effect(A,aa(B,heap_Time_Heap(A),G3,R),H4,H6,R6,N22)
               => ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_306_effect__bindI,axiom,
    ! [B: $tType,A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat,G3: fun(A,heap_Time_Heap(B)),H6: heap_ext(product_unit),R6: B,N2: nat] :
      ( heap_Time_effect(A,F3,H,H5,R3,N)
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),G3,R3),H5,H6,R6,N2)
       => heap_Time_effect(B,heap_Time_bind(A,B,F3,G3),H,H6,R6,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)) ) ) ).

% effect_bindI
tff(fact_307_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).

% verit_comp_simplify1(2)
tff(fact_308_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% verit_la_disequality
tff(fact_309_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).

% verit_comp_simplify1(1)
tff(fact_310_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A3: A] :
        ? [B4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B4))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),A3)) ) ) ).

% ex_gt_or_lt
tff(fact_311_in__mono,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,member(A,X),A6))
       => pp(aa(set(A),bool,member(A,X),B5)) ) ) ).

% in_mono
tff(fact_312_subsetD,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C3: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,member(A,C3),A6))
       => pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% subsetD
tff(fact_313_psubsetE,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6)) ) ) ).

% psubsetE
tff(fact_314_equalityE,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( A6 = B5 )
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6)) ) ) ).

% equalityE
tff(fact_315_subset__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A6))
         => pp(aa(set(A),bool,member(A,X4),B5)) ) ) ).

% subset_eq
tff(fact_316_equalityD1,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( A6 = B5 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ).

% equalityD1
tff(fact_317_equalityD2,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( A6 = B5 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6)) ) ).

% equalityD2
tff(fact_318_psubset__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
        & ( A6 != B5 ) ) ) ).

% psubset_eq
tff(fact_319_subset__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
    <=> ! [T6: A] :
          ( pp(aa(set(A),bool,member(A,T6),A6))
         => pp(aa(set(A),bool,member(A,T6),B5)) ) ) ).

% subset_iff
tff(fact_320_subset__refl,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),A6)) ).

% subset_refl
tff(fact_321_Collect__mono,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
         => pp(aa(A,bool,Q2,X3)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),P2)),aa(fun(A,bool),set(A),collect(A),Q2))) ) ).

% Collect_mono
tff(fact_322_subset__trans,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4)) ) ) ).

% subset_trans
tff(fact_323_set__eq__subset,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( A6 = B5 )
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6)) ) ) ).

% set_eq_subset
tff(fact_324_Collect__mono__iff,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),P2)),aa(fun(A,bool),set(A),collect(A),Q2)))
    <=> ! [X4: A] :
          ( pp(aa(A,bool,P2,X4))
         => pp(aa(A,bool,Q2,X4)) ) ) ).

% Collect_mono_iff
tff(fact_325_psubset__imp__subset,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ).

% psubset_imp_subset
tff(fact_326_psubset__subset__trans,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),C4)) ) ) ).

% psubset_subset_trans
tff(fact_327_subset__not__subset__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
        & ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6)) ) ) ).

% subset_not_subset_eq
tff(fact_328_subset__psubset__trans,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B5),C4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),C4)) ) ) ).

% subset_psubset_trans
tff(fact_329_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
        | ( A6 = B5 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_330_Collect__subset,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2))),A6)) ).

% Collect_subset
tff(fact_331_less__eq__set__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B5))) ) ).

% less_eq_set_def
tff(fact_332_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z4))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4)) ) ) ) ).

% le_sup_iff
tff(fact_333_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% sup.bounded_iff
tff(fact_334_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_left
tff(fact_335_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_right
tff(fact_336_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_left
tff(fact_337_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_right
tff(fact_338_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_less_le_mono
tff(fact_339_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_le_less_mono
tff(fact_340_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_341_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_342_sup_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),A3) = A3 ) ).

% sup.idem
tff(fact_343_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
        <=> ( B2 = C3 ) ) ) ).

% add_right_cancel
tff(fact_344_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add_left_cancel
tff(fact_345_sup__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F3),G3),X) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,X)),aa(A,B,G3,X)) ) ).

% sup_apply
tff(fact_346_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.right_idem
tff(fact_347_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).

% sup_left_idem
tff(fact_348_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).

% sup.left_idem
tff(fact_349_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),X) = X ) ).

% sup_idem
tff(fact_350_mod__or__dist,axiom,
    ! [P2: assn,Q2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2)),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H))
        | pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),H)) ) ) ).

% mod_or_dist
tff(fact_351_predicate1I,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
         => pp(aa(A,bool,Q2,X3)) )
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Q2)) ) ).

% predicate1I
tff(fact_352_rev__predicate1D,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,Q2: fun(A,bool)] :
      ( pp(aa(A,bool,P2,X))
     => ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Q2))
       => pp(aa(A,bool,Q2,X)) ) ) ).

% rev_predicate1D
tff(fact_353_predicate1D,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool),X: A] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Q2))
     => ( pp(aa(A,bool,P2,X))
       => pp(aa(A,bool,Q2,X)) ) ) ).

% predicate1D
tff(fact_354_sup__set__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B5))) ).

% sup_set_def
tff(fact_355_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
         => ( B2 = C3 ) ) ) ).

% add_right_imp_eq
tff(fact_356_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
         => ( B2 = C3 ) ) ) ).

% add_left_imp_eq
tff(fact_357_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_358_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_359_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
        <=> ( B2 = C3 ) ) ) ).

% add.right_cancel
tff(fact_360_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add.left_cancel
tff(fact_361_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.assoc
tff(fact_362_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: A,K2: A,B2: A,A3: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_363_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: A,K2: A,A3: A,B2: A] :
          ( ( A6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_364_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( ( I2 = J2 )
            & ( K2 = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_365_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X5: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F3),G3),X5) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,X5)),aa(A,B,G3,X5)) ) ).

% sup_fun_def
tff(fact_366_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z4)) ) ).

% sup_left_commute
tff(fact_367_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)) ) ).

% sup.left_commute
tff(fact_368_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).

% sup_commute
tff(fact_369_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),A3) ) ).

% sup.commute
tff(fact_370_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z4) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)) ) ).

% sup_assoc
tff(fact_371_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)) ) ).

% sup.assoc
tff(fact_372_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).

% inf_sup_aci(5)
tff(fact_373_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z4) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)) ) ).

% inf_sup_aci(6)
tff(fact_374_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z4)) ) ).

% inf_sup_aci(7)
tff(fact_375_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).

% inf_sup_aci(8)
tff(fact_376_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_377_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_378_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ? [C5: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C5) ) ) ).

% le_iff_add
tff(fact_379_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ).

% add_right_mono
tff(fact_380_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ~ ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ) ).

% less_eqE
tff(fact_381_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% add_left_mono
tff(fact_382_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_mono
tff(fact_383_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_384_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( ( I2 = J2 )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_385_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J2))
            & ( K2 = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_386_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_387_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_388_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ).

% add_strict_right_mono
tff(fact_389_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% add_strict_left_mono
tff(fact_390_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_strict_mono
tff(fact_391_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J2))
            & ( K2 = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_392_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( ( I2 = J2 )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_393_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_394_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI2
tff(fact_395_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI1
tff(fact_396_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_397_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb_iff1
tff(fact_398_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded2
tff(fact_399_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded1
tff(fact_400_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.order_iff
tff(fact_401_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3)) ) ) ) ).

% sup.boundedI
tff(fact_402_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% sup.boundedE
tff(fact_403_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_404_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_405_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_406_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb1
tff(fact_407_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)))
         => ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)))
           => ( ! [X3: A,Y3: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3)),X3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_408_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% sup.orderI
tff(fact_409_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.orderE
tff(fact_410_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_411_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)),X)) ) ) ) ).

% sup_least
tff(fact_412_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3))) ) ) ) ).

% sup_mono
tff(fact_413_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,D3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ) ).

% sup.mono
tff(fact_414_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI2
tff(fact_415_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI1
tff(fact_416_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge2
tff(fact_417_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge1
tff(fact_418_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)) ) ) ) ).

% le_supI
tff(fact_419_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).

% le_supE
tff(fact_420_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(3)
tff(fact_421_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(4)
tff(fact_422_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.strict_coboundedI2
tff(fact_423_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.strict_coboundedI1
tff(fact_424_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_425_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% sup.strict_boundedE
tff(fact_426_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_427_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb3
tff(fact_428_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% less_supI2
tff(fact_429_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% less_supI1
tff(fact_430_pred__subset__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),R2)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R2),S)) ) ).

% pred_subset_eq
tff(fact_431_timeFrame_Oelims,axiom,
    ! [A: $tType,X: nat,Xa2: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,X,Xa2) = Y )
     => ( ! [R: A,H2: heap_ext(product_unit),N6: nat] :
            ( ( Xa2 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N6))) )
           => ( Y != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N6)))) ) )
       => ~ ( ( Xa2 = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
           => ( Y != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ) ).

% timeFrame.elims
tff(fact_432_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_ak(A,fun(A,bool)),aTP_Lamp_al(A,fun(A,bool))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_433_sup__option__def,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X),Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_an(option(A),fun(option(A),fun(A,option(A))),X),Y),X) ) ).

% sup_option_def
tff(fact_434_sup__Un__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),R2)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R2),S))) ) ).

% sup_Un_eq
tff(fact_435_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),S)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),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))),R2),S))) ) ).

% sup_Un_eq2
tff(fact_436_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),S)))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R2),S)) ) ).

% pred_subset_eq2
tff(fact_437_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Xa3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3)),R2))
        <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3)),S)) )
    <=> ( R2 = S ) ) ).

% pred_equals_eq2
tff(fact_438_execute__guard_I2_J,axiom,
    ! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( pp(aa(heap_ext(product_unit),bool,P2,H))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_guard(A,P2,F3)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H)) ) ) ).

% execute_guard(2)
tff(fact_439_some__opt__eq__trivial,axiom,
    ! [A: $tType,X: A] : eps_Opt(A,aTP_Lamp_ap(A,fun(A,bool),X)) = aa(A,option(A),some(A),X) ).

% some_opt_eq_trivial
tff(fact_440_raise__bind,axiom,
    ! [B: $tType,A: $tType,E3: list(char),F3: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,heap_Time_raise(B,E3),F3) = heap_Time_raise(A,E3) ).

% raise_bind
tff(fact_441_predicate2I,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),Q2: fun(A,fun(B,bool))] :
      ( ! [X3: A,Y3: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X3),Y3))
         => pp(aa(B,bool,aa(A,fun(B,bool),Q2,X3),Y3)) )
     => pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P2),Q2)) ) ).

% predicate2I
tff(fact_442_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y4: A] : X = aa(A,option(A),some(A),Y4) ) ).

% not_None_eq
tff(fact_443_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y4: A] : X != aa(A,option(A),some(A),Y4)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_444_some__opt__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : eps_Opt(A,aa(A,fun(A,bool),fequal(A),X)) = aa(A,option(A),some(A),X) ).

% some_opt_sym_eq_trivial
tff(fact_445_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),none(A)),X)) ) ).

% less_eq_option_None_code
tff(fact_446_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : ~ pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),X),none(A))) ) ).

% less_option_None
tff(fact_447_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_448_sup__None__2,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [X: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X),none(A)) = X ) ).

% sup_None_2
tff(fact_449_some__opt__false__trivial,axiom,
    ! [A: $tType] : eps_Opt(A,aTP_Lamp_aq(A,bool)) = none(A) ).

% some_opt_false_trivial
tff(fact_450_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V2: option(product_prod(A,B))] :
      ( ! [X4: A,Y4: B] : V2 != 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),X4),Y4))
    <=> ( V2 = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_451_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),X)),none(A))) ) ).

% less_eq_option_Some_None
tff(fact_452_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),aa(A,option(A),some(A),X))) ) ).

% less_option_None_Some_code
tff(fact_453_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),X: A,Y: B,Q2: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X),Y))
     => ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P2),Q2))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q2,X),Y)) ) ) ).

% rev_predicate2D
tff(fact_454_predicate2D,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Q2: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P2),Q2))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q2,X),Y)) ) ) ).

% predicate2D
tff(fact_455_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: fun(A,B)] : case_option(B,A,F1,F22,none(A)) = F1 ).

% option.simps(4)
tff(fact_456_execute__guard_I1_J,axiom,
    ! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ~ pp(aa(heap_ext(product_unit),bool,P2,H))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_guard(A,P2,F3)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_guard(1)
tff(fact_457_execute__raise,axiom,
    ! [A: $tType,S2: list(char),X5: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_raise(A,S2)),X5) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% execute_raise
tff(fact_458_option_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,B),Option: option(A)] : aa(B,C,H,case_option(B,A,F1,F22,Option)) = case_option(C,A,aa(B,C,H,F1),aa(fun(A,B),fun(A,C),aTP_Lamp_ar(fun(B,C),fun(fun(A,B),fun(A,C)),H),F22),Option) ).

% option.case_distrib
tff(fact_459_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X2: A] : none(A) != aa(A,option(A),some(A),X2) ).

% option.distinct(1)
tff(fact_460_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_461_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X22: A] : Y != aa(A,option(A),some(A),X22) ) ).

% option.exhaust
tff(fact_462_split__option__ex,axiom,
    ! [A: $tType,P2: fun(option(A),bool)] :
      ( ? [X_12: option(A)] : pp(aa(option(A),bool,P2,X_12))
    <=> ( pp(aa(option(A),bool,P2,none(A)))
        | ? [X4: A] : pp(aa(option(A),bool,P2,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_ex
tff(fact_463_split__option__all,axiom,
    ! [A: $tType,P2: fun(option(A),bool)] :
      ( ! [X_12: option(A)] : pp(aa(option(A),bool,P2,X_12))
    <=> ( pp(aa(option(A),bool,P2,none(A)))
        & ! [X4: A] : pp(aa(option(A),bool,P2,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_all
tff(fact_464_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P2: fun(option(A),fun(option(B),bool)),Y: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) )
     => ( ( ( Y = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) )
       => ( ! [A5: A,B4: B] :
              ( ( X = aa(A,option(A),some(A),A5) )
             => ( ( Y = aa(B,option(B),some(B),B4) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_465_timeFrame_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat] : heap_Time_timeFrame(A,N,none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% timeFrame.simps(2)
tff(fact_466_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),X),none(A)))
         => ( X = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_467_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),none(A)),X)) ) ).

% less_eq_option_None
tff(fact_468_option_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: fun(A,B),X2: A] : case_option(B,A,F1,F22,aa(A,option(A),some(A),X2)) = aa(A,B,F22,X2) ).

% option.simps(5)
tff(fact_469_the__default_Osimps_I2_J,axiom,
    ! [A: $tType,X: A] : the_default(A,X,none(A)) = X ).

% the_default.simps(2)
tff(fact_470_Eps__Opt__eq__Some__implies,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A] :
      ( ( eps_Opt(A,P2) = aa(A,option(A),some(A),X) )
     => pp(aa(A,bool,P2,X)) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_471_Eps__Opt__eq__Some,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A] :
      ( ! [X7: A] :
          ( pp(aa(A,bool,P2,X))
         => ( pp(aa(A,bool,P2,X7))
           => ( X7 = X ) ) )
     => ( ( eps_Opt(A,P2) = aa(A,option(A),some(A),X) )
      <=> pp(aa(A,bool,P2,X)) ) ) ).

% Eps_Opt_eq_Some
tff(fact_472_execute__bind_I2_J,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),G3: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F3,G3)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_bind(2)
tff(fact_473_success__def,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F3,H)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% success_def
tff(fact_474_successI,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
     => heap_Time_success(A,F3,H) ) ).

% successI
tff(fact_475_success__guardI,axiom,
    ! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( pp(aa(heap_ext(product_unit),bool,P2,H))
     => heap_Time_success(A,heap_Time_guard(A,P2,F3),H) ) ).

% success_guardI
tff(fact_476_effect__raiseE,axiom,
    ! [A: $tType,X: list(char),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] : ~ heap_Time_effect(A,heap_Time_raise(A,X),H,H5,R3,N) ).

% effect_raiseE
tff(fact_477_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),X))
         => ? [Z3: A] : X = aa(A,option(A),some(A),Z3) ) ) ).

% less_option_None_is_Some
tff(fact_478_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),aa(A,option(A),some(A),X))) ) ).

% less_option_None_Some
tff(fact_479_subrelI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X3: A,Y3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R3))
         => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),S2)) )
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S2)) ) ).

% subrelI
tff(fact_480_Heap__cases,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( ! [X3: A,H4: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X3),H4))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% Heap_cases
tff(fact_481_timeFrame_Ocases,axiom,
    ! [A: $tType,X: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( ! [N4: nat,R: A,H2: heap_ext(product_unit),N6: nat] : X != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N4),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N6))))
     => ~ ! [N4: nat] : X != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N4),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).

% timeFrame.cases
tff(fact_482_sup1CI,axiom,
    ! [A: $tType,B5: fun(A,bool),X: A,A6: fun(A,bool)] :
      ( ( ~ pp(aa(A,bool,B5,X))
       => pp(aa(A,bool,A6,X)) )
     => pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A6),B5),X)) ) ).

% sup1CI
tff(fact_483_sup2CI,axiom,
    ! [A: $tType,B: $tType,B5: fun(A,fun(B,bool)),X: A,Y: B,A6: fun(A,fun(B,bool))] :
      ( ( ~ pp(aa(B,bool,aa(A,fun(B,bool),B5,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),A6,X),Y)) )
     => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A6),B5),X),Y)) ) ).

% sup2CI
tff(fact_484_reflclp__idemp,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A))),fequal(A)) = aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)) ).

% reflclp_idemp
tff(fact_485_timeFrame_Opelims,axiom,
    ! [A: $tType,X: nat,Xa2: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X),Xa2)))
       => ( ! [R: A,H2: heap_ext(product_unit),N6: nat] :
              ( ( Xa2 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N6))) )
             => ( ( Y = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N6)))) )
               => ~ pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N6)))))) ) )
         => ~ ( ( Xa2 = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
             => ( ( Y = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
               => ~ pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_486_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P2: bool,Q2: fun(A,bool),X: option(A),R2: fun(B,bool),F3: B,G3: fun(A,B)] :
      ( pp(case_option(bool,A,P2,Q2,X))
     => ( ( pp(P2)
         => pp(aa(B,bool,R2,F3)) )
       => ( ! [Q4: A] :
              ( pp(aa(A,bool,Q2,Q4))
             => pp(aa(B,bool,R2,aa(A,B,G3,Q4))) )
         => pp(aa(B,bool,R2,case_option(B,A,F3,G3,X))) ) ) ) ).

% disjE_realizer2
tff(fact_487_eq__subset,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool))] : pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aTP_Lamp_as(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P2))) ).

% eq_subset
tff(fact_488_set__to__map__def,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(B,A)),K2: B] : aa(B,option(A),set_to_map(B,A,S),K2) = eps_Opt(A,aa(B,fun(A,bool),aTP_Lamp_at(set(product_prod(B,A)),fun(B,fun(A,bool)),S),K2)) ).

% set_to_map_def
tff(fact_489_sup1E,axiom,
    ! [A: $tType,A6: fun(A,bool),B5: fun(A,bool),X: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A6),B5),X))
     => ( ~ pp(aa(A,bool,A6,X))
       => pp(aa(A,bool,B5,X)) ) ) ).

% sup1E
tff(fact_490_sup1I1,axiom,
    ! [A: $tType,A6: fun(A,bool),X: A,B5: fun(A,bool)] :
      ( pp(aa(A,bool,A6,X))
     => pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A6),B5),X)) ) ).

% sup1I1
tff(fact_491_sup1I2,axiom,
    ! [A: $tType,B5: fun(A,bool),X: A,A6: fun(A,bool)] :
      ( pp(aa(A,bool,B5,X))
     => pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),sup_sup(fun(A,bool)),A6),B5),X)) ) ).

% sup1I2
tff(fact_492_guard__def,axiom,
    ! [A: $tType,P2: fun(heap_ext(product_unit),bool),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_guard(A,P2,F3) = heap_Time_Heap2(A,aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_au(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),P2),F3)) ).

% guard_def
tff(fact_493_Heap_Oinject,axiom,
    ! [A: $tType,X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Ya: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( ( heap_Time_Heap2(A,X) = heap_Time_Heap2(A,Ya) )
    <=> ( X = Ya ) ) ).

% Heap.inject
tff(fact_494_Heap__execute,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A)] : heap_Time_Heap2(A,heap_Time_execute(A,F3)) = F3 ).

% Heap_execute
tff(fact_495_Heap_Oexhaust,axiom,
    ! [A: $tType,Y: heap_Time_Heap(A)] :
      ~ ! [X3: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : Y != heap_Time_Heap2(A,X3) ).

% Heap.exhaust
tff(fact_496_execute_Osimps,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_execute(A,heap_Time_Heap2(A,F3)) = F3 ).

% execute.simps
tff(fact_497_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(case_option(bool,A,fFalse,aTP_Lamp_av(A,bool),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_498_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(case_option(bool,A,fTrue,aTP_Lamp_aq(A,bool),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_499_case__optionE,axiom,
    ! [A: $tType,P2: bool,Q2: fun(A,bool),X: option(A)] :
      ( pp(case_option(bool,A,P2,Q2,X))
     => ( ( ( X = none(A) )
         => ~ pp(P2) )
       => ~ ! [Y3: A] :
              ( ( X = aa(A,option(A),some(A),Y3) )
             => ~ pp(aa(A,bool,Q2,Y3)) ) ) ) ).

% case_optionE
tff(fact_500_boolean__algebra__cancel_Osup2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B5: A,K2: A,B2: A,A3: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B5) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_501_boolean__algebra__cancel_Osup1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: A,K2: A,A3: A,B2: A] :
          ( ( A6 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K2),A3) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_502_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),X),Y))
        <=> pp(case_option(bool,A,fTrue,aTP_Lamp_aw(option(A),fun(A,bool),Y),X)) ) ) ).

% less_eq_option_def
tff(fact_503_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: option(A),Y: option(A)] :
          ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),X),Y))
        <=> pp(case_option(bool,A,fFalse,aTP_Lamp_ay(option(A),fun(A,bool),X),Y)) ) ) ).

% less_option_def
tff(fact_504_raise__def,axiom,
    ! [A: $tType,S2: list(char)] : heap_Time_raise(A,S2) = heap_Time_Heap2(A,aTP_Lamp_az(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).

% raise_def
tff(fact_505_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Q2: fun(A,fun(B,bool)),R2: bool,X: A,Y: B] :
      ( ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P2),Q2))
        & pp(R2) )
     => ( pp(R2)
        & ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X),Y))
         => pp(aa(B,bool,aa(A,fun(B,bool),Q2,X),Y)) ) ) ) ).

% predicate2D_conj
tff(fact_506_sup2I2,axiom,
    ! [A: $tType,B: $tType,B5: fun(A,fun(B,bool)),X: A,Y: B,A6: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),B5,X),Y))
     => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A6),B5),X),Y)) ) ).

% sup2I2
tff(fact_507_sup2I1,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool)),X: A,Y: B,B5: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),A6,X),Y))
     => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A6),B5),X),Y)) ) ).

% sup2I1
tff(fact_508_sup2E,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),A6),B5),X),Y))
     => ( ~ pp(aa(B,bool,aa(A,fun(B,bool),A6,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),B5,X),Y)) ) ) ).

% sup2E
tff(fact_509_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,bool)),R22: fun(A,fun(A,bool))] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R1),R22))
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),accp(A,R22)),accp(A,R1))) ) ).

% accp_subset
tff(fact_510_accp__subset__induct,axiom,
    ! [A: $tType,D5: fun(A,bool),R2: fun(A,fun(A,bool)),X: A,P2: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),D5),accp(A,R2)))
     => ( ! [X3: A,Z3: A] :
            ( pp(aa(A,bool,D5,X3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,Z3),X3))
             => pp(aa(A,bool,D5,Z3)) ) )
       => ( pp(aa(A,bool,D5,X))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,D5,X3))
               => ( ! [Z6: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),R2,Z6),X3))
                     => pp(aa(A,bool,P2,Z6)) )
                 => pp(aa(A,bool,P2,X3)) ) )
           => pp(aa(A,bool,P2,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_511_execute__assert_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,H: heap_ext(product_unit)] :
      ( ~ pp(aa(A,bool,P2,X))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P2,X)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_assert(2)
tff(fact_512_mod__h__bot__iff_I7_J,axiom,
    ! [P2: assn,Q2: assn,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
        | pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_iff(7)
tff(fact_513_combine__options__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A)] : combine_options(A,F3,X,Y) = case_option(option(A),A,Y,aa(option(A),fun(A,option(A)),aTP_Lamp_bb(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F3),Y),X) ).

% combine_options_def
tff(fact_514_arg__max__nat__le,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P2,X))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,lattices_ord_arg_max(A,nat,F3,P2)))) ) ) ).

% arg_max_nat_le
tff(fact_515_arg__max__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P2,K2))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => ( pp(aa(A,bool,P2,lattices_ord_arg_max(A,nat,F3,P2)))
          & ! [Y5: A] :
              ( pp(aa(A,bool,P2,Y5))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y5)),aa(A,nat,F3,lattices_ord_arg_max(A,nat,F3,P2)))) ) ) ) ) ).

% arg_max_nat_lemma
tff(fact_516_reflcl__set__eq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)),fequal(A)),X5),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),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))),R3),id2(A)))) ) ).

% reflcl_set_eq
tff(fact_517_in__measure,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measure(A,F3)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y))) ) ).

% in_measure
tff(fact_518_wait__bind__decon,axiom,
    ! [A: $tType,P2: assn,M: heap_Time_Heap(A),Q2: fun(A,assn),N: nat] :
      ( hoare_hoare_triple(A,P2,M,Q2)
     => hoare_hoare_triple(A,P2,heap_Time_bind(product_unit,A,heap_Time_wait(N),aTP_Lamp_bd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),Q2) ) ).

% wait_bind_decon
tff(fact_519_bot__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( bot(C)
     => ! [X: D] : aa(D,C,bot_bot(fun(D,C)),X) = bot_bot(C) ) ).

% bot_apply
tff(fact_520_empty__Collect__eq,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ( bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),P2) )
    <=> ! [X4: A] : ~ pp(aa(A,bool,P2,X4)) ) ).

% empty_Collect_eq
tff(fact_521_Collect__empty__eq,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ( aa(fun(A,bool),set(A),collect(A),P2) = bot_bot(set(A)) )
    <=> ! [X4: A] : ~ pp(aa(A,bool,P2,X4)) ) ).

% Collect_empty_eq
tff(fact_522_all__not__in__conv,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ! [X4: A] : ~ pp(aa(set(A),bool,member(A,X4),A6))
    <=> ( A6 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_523_empty__iff,axiom,
    ! [A: $tType,C3: A] : ~ pp(aa(set(A),bool,member(A,C3),bot_bot(set(A)))) ).

% empty_iff
tff(fact_524_subset__empty,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),bot_bot(set(A))))
    <=> ( A6 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_525_empty__subsetI,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),bot_bot(set(A))),A6)) ).

% empty_subsetI
tff(fact_526_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).

% sup_bot_left
tff(fact_527_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% sup_bot_right
tff(fact_528_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_529_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = bot_bot(A) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_530_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = bot_bot(A) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_531_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A3) = A3 ) ).

% sup_bot.left_neutral
tff(fact_532_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_533_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),bot_bot(A)) = A3 ) ).

% sup_bot.right_neutral
tff(fact_534_Un__empty,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = bot_bot(set(A)) )
    <=> ( ( A6 = bot_bot(set(A)) )
        & ( B5 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_535_pair__in__Id__conv,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
    <=> ( A3 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_536_IdI,axiom,
    ! [A: $tType,A3: A] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id2(A))) ).

% IdI
tff(fact_537_combine__options__simps_I3_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),A3: A,B2: A] : combine_options(A,F3,aa(A,option(A),some(A),A3),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F3,A3),B2)) ).

% combine_options_simps(3)
tff(fact_538_combine__options__simps_I1_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Y: option(A)] : combine_options(A,F3,none(A),Y) = Y ).

% combine_options_simps(1)
tff(fact_539_combine__options__simps_I2_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A)] : combine_options(A,F3,X,none(A)) = X ).

% combine_options_simps(2)
tff(fact_540_bijective__Id,axiom,
    ! [A: $tType] : bijective(A,A,id2(A)) ).

% bijective_Id
tff(fact_541_memb__imp__not__empty,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),S))
     => ( S != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_542_set__notEmptyE,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ~ ! [X3: A] : ~ pp(aa(set(A),bool,member(A,X3),S)) ) ).

% set_notEmptyE
tff(fact_543_combine__options__left__commute,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Y: option(A),X: option(A),Z4: option(A)] :
      ( ! [X3: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F3,X3),Y3) = aa(A,A,aa(A,fun(A,A),F3,Y3),X3)
     => ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Z3) = aa(A,A,aa(A,fun(A,A),F3,X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))
       => ( combine_options(A,F3,Y,combine_options(A,F3,X,Z4)) = combine_options(A,F3,X,combine_options(A,F3,Y,Z4)) ) ) ) ).

% combine_options_left_commute
tff(fact_544_combine__options__commute,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A)] :
      ( ! [X3: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F3,X3),Y3) = aa(A,A,aa(A,fun(A,A),F3,Y3),X3)
     => ( combine_options(A,F3,X,Y) = combine_options(A,F3,Y,X) ) ) ).

% combine_options_commute
tff(fact_545_combine__options__assoc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A),Z4: option(A)] :
      ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Z3) = aa(A,A,aa(A,fun(A,A),F3,X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))
     => ( combine_options(A,F3,combine_options(A,F3,X,Y),Z4) = combine_options(A,F3,X,combine_options(A,F3,Y,Z4)) ) ) ).

% combine_options_assoc
tff(fact_546_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X5: A] : aa(A,B,bot_bot(fun(A,B)),X5) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_547_ex__in__conv,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ? [X4: A] : pp(aa(set(A),bool,member(A,X4),A6))
    <=> ( A6 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_548_equals0I,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ! [Y3: A] : ~ pp(aa(set(A),bool,member(A,Y3),A6))
     => ( A6 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_549_equals0D,axiom,
    ! [A: $tType,A6: set(A),A3: A] :
      ( ( A6 = bot_bot(set(A)) )
     => ~ pp(aa(set(A),bool,member(A,A3),A6)) ) ).

% equals0D
tff(fact_550_emptyE,axiom,
    ! [A: $tType,A3: A] : ~ pp(aa(set(A),bool,member(A,A3),bot_bot(set(A)))) ).

% emptyE
tff(fact_551_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aq(A,bool)) ).

% Set.empty_def
tff(fact_552_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A3)) ) ).

% bot.extremum
tff(fact_553_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_554_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_555_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),bot_bot(A))) ) ).

% bot.extremum_strict
tff(fact_556_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A3)) ) ) ).

% bot.not_eq_extremum
tff(fact_557_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_558_IdE,axiom,
    ! [A: $tType,P3: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),id2(A)))
     => ~ ! [X3: A] : P3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).

% IdE
tff(fact_559_BNF__Greatest__Fixpoint_OIdD,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
     => ( A3 = B2 ) ) ).

% BNF_Greatest_Fixpoint.IdD
tff(fact_560_subset__emptyI,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ! [X3: A] : ~ pp(aa(set(A),bool,member(A,X3),A6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),bot_bot(set(A)))) ) ).

% subset_emptyI
tff(fact_561_Un__empty__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B5) = B5 ).

% Un_empty_left
tff(fact_562_Un__empty__right,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),bot_bot(set(A))) = A6 ).

% Un_empty_right
tff(fact_563_not__psubset__empty,axiom,
    ! [A: $tType,A6: set(A)] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),bot_bot(set(A)))) ).

% not_psubset_empty
tff(fact_564_arg__max__natI,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P2,K2))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => pp(aa(A,bool,P2,lattices_ord_arg_max(A,nat,F3,P2))) ) ) ).

% arg_max_natI
tff(fact_565_mod__h__bot__indep,axiom,
    ! [P2: assn,H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),bot_bot(set(nat))))) ) ).

% mod_h_bot_indep
tff(fact_566_assert__cong,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),P4: fun(A,bool),F3: fun(A,heap_Time_Heap(B)),F5: fun(A,heap_Time_Heap(B)),X: A] :
      ( ( P2 = P4 )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P4,X3))
           => ( aa(A,heap_Time_Heap(B),F3,X3) = aa(A,heap_Time_Heap(B),F5,X3) ) )
       => ( heap_Time_bind(A,B,heap_Time_assert(A,P2,X),F3) = heap_Time_bind(A,B,heap_Time_assert(A,P4,X),F5) ) ) ) ).

% assert_cong
tff(fact_567_success__assertI,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,H: heap_ext(product_unit)] :
      ( pp(aa(A,bool,P2,X))
     => heap_Time_success(A,heap_Time_assert(A,P2,X),H) ) ).

% success_assertI
tff(fact_568_arg__max__equality,axiom,
    ! [A: $tType,C: $tType] :
      ( order(A)
     => ! [P2: fun(C,bool),K2: C,F3: fun(C,A)] :
          ( pp(aa(C,bool,P2,K2))
         => ( ! [X3: C] :
                ( pp(aa(C,bool,P2,X3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X3)),aa(C,A,F3,K2))) )
           => ( aa(C,A,F3,lattices_ord_arg_max(C,A,F3,P2)) = aa(C,A,F3,K2) ) ) ) ) ).

% arg_max_equality
tff(fact_569_arg__maxI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P2: fun(A,bool),X: A,F3: fun(A,B),Q2: fun(A,bool)] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y3))) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P2,X3))
                 => ( ! [Y5: A] :
                        ( pp(aa(A,bool,P2,Y5))
                       => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y5))) )
                   => pp(aa(A,bool,Q2,X3)) ) )
             => pp(aa(A,bool,Q2,lattices_ord_arg_max(A,B,F3,P2))) ) ) ) ) ).

% arg_maxI
tff(fact_570_arg__max__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: fun(B,A),S: set(B)] : lattic1883929316492267755max_on(B,A,F3,S) = lattices_ord_arg_max(B,A,F3,aTP_Lamp_be(set(B),fun(B,bool),S)) ) ).

% arg_max_on_def
tff(fact_571_mod__h__bot__iff_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),X: A,H: heap_ext(product_unit)] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(sngr_assn(A,P3,X)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ).

% mod_h_bot_iff(3)
tff(fact_572_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_bf(A,fun(A,bool)),aTP_Lamp_bg(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_573_mod__h__bot__iff_I4_J,axiom,
    ! [B: $tType] :
      ( heap(B)
     => ! [Q5: array(B),Y: list(B),H: heap_ext(product_unit)] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(snga_assn(B,Q5,Y)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ).

% mod_h_bot_iff(4)
tff(fact_574_execute__assert_I1_J,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,H: heap_ext(product_unit)] :
      ( pp(aa(A,bool,P2,X))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P2,X)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ) ).

% execute_assert(1)
tff(fact_575_mod__h__bot__iff_I8_J,axiom,
    ! [C: $tType,R2: fun(C,assn),H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(C,R2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> ? [X4: C] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(C,assn,R2,X4)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ).

% mod_h_bot_iff(8)
tff(fact_576_the__dflt__None__empty,axiom,
    ! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).

% the_dflt_None_empty
tff(fact_577_the__dflt__None__nonempty,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( dflt_None_set(A,S) = aa(set(A),option(set(A)),some(set(A)),S) ) ) ).

% the_dflt_None_nonempty
tff(fact_578_the__dflt__None__set,axiom,
    ! [A: $tType,X: set(A)] : the_default(set(A),bot_bot(set(A)),dflt_None_set(A,X)) = X ).

% the_dflt_None_set
tff(fact_579_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_bh(A,fun(A,bool)),aTP_Lamp_bi(A,fun(A,bool)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_580_false__rule,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),Q2: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),C3,Q2) ).

% false_rule
tff(fact_581_ex__assn__const,axiom,
    ! [A: $tType,C3: assn] : ex_assn(A,aTP_Lamp_bj(assn,fun(A,assn),C3)) = C3 ).

% ex_assn_const
tff(fact_582_bijective__Empty,axiom,
    ! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).

% bijective_Empty
tff(fact_583_set__to__map__empty,axiom,
    ! [A: $tType,B: $tType,X5: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X5) = none(B) ).

% set_to_map_empty
tff(fact_584_mod__ex__dist,axiom,
    ! [A: $tType,P2: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(A,P2)),H))
    <=> ? [X4: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P2,X4)),H)) ) ).

% mod_ex_dist
tff(fact_585_ordering__top_Oextremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),Top)) ) ).

% ordering_top.extremum
tff(fact_586_ordering__top_Oextremum__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Top),A3)) ) ).

% ordering_top.extremum_strict
tff(fact_587_ordering__top_Oextremum__unique,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
      <=> ( A3 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_588_ordering__top_Onot__eq__extremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( ( A3 != Top )
      <=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),Top)) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_589_ordering__top_Oextremum__uniqueI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
       => ( A3 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_590_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),bot_bot(set(product_prod(A,B))))) ) ).

% bot_empty_eq2
tff(fact_591_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) ).

% bot_set_def
tff(fact_592_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_593_mod__false,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(bot_bot(assn)),H)) ).

% mod_false
tff(fact_594_less__by__empty,axiom,
    ! [A: $tType,A6: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
      ( ( A6 = bot_bot(set(product_prod(A,A))) )
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A6),B5)) ) ).

% less_by_empty
tff(fact_595_mod__exE,axiom,
    ! [A: $tType,P2: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(A,P2)),H))
     => ~ ! [X3: A] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P2,X3)),H)) ) ).

% mod_exE
tff(fact_596_mod__exI,axiom,
    ! [A: $tType,P2: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( ? [X5: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P2,X5)),H))
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(ex_assn(A,P2)),H)) ) ).

% mod_exI
tff(fact_597_ex__one__point__gen,axiom,
    ! [A: $tType,P2: fun(A,assn),V2: A] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat)),X3: A] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,P2,X3)),H2))
         => ( X3 = V2 ) )
     => ( ex_assn(A,P2) = aa(A,assn,P2,V2) ) ) ).

% ex_one_point_gen
tff(fact_598_post__exI__rule,axiom,
    ! [B: $tType,A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,fun(B,assn)),X: B] :
      ( hoare_hoare_triple(A,P2,C3,aa(B,fun(A,assn),aTP_Lamp_bk(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Q2),X))
     => hoare_hoare_triple(A,P2,C3,aTP_Lamp_bl(fun(A,fun(B,assn)),fun(A,assn),Q2)) ) ).

% post_exI_rule
tff(fact_599_norm__pre__ex__rule,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,assn),F3: heap_Time_Heap(B),Q2: fun(B,assn)] :
      ( ! [X3: A] : hoare_hoare_triple(B,aa(A,assn,P2,X3),F3,Q2)
     => hoare_hoare_triple(B,ex_assn(A,P2),F3,Q2) ) ).

% norm_pre_ex_rule
tff(fact_600_ex__join__or,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: fun(A,assn)] : ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_bm(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P2),Q2)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_bn(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P2),Q2)) ).

% ex_join_or
tff(fact_601_ex__distrib__or,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_bo(fun(A,assn),fun(assn,fun(A,assn)),P2),Q2)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,P2)),Q2) ).

% ex_distrib_or
tff(fact_602_raise__rule,axiom,
    ! [A: $tType,S2: list(char),Q2: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),heap_Time_raise(A,S2),Q2) ).

% raise_rule
tff(fact_603_raise__iff,axiom,
    ! [A: $tType,P2: assn,S2: list(char),Q2: fun(A,assn)] :
      ( hoare_hoare_triple(A,P2,heap_Time_raise(A,S2),Q2)
    <=> ( P2 = bot_bot(assn) ) ) ).

% raise_iff
tff(fact_604_set__to__map__empty__iff_I1_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ! [X4: A] : aa(A,option(B),set_to_map(A,B,S),X4) = none(B)
    <=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(1)
tff(fact_605_effect__assertI,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,H5: heap_ext(product_unit),H: heap_ext(product_unit),R3: A,N: nat] :
      ( pp(aa(A,bool,P2,X))
     => ( ( H5 = H )
       => ( ( R3 = X )
         => ( ( N = one_one(nat) )
           => heap_Time_effect(A,heap_Time_assert(A,P2,X),H,H5,R3,N) ) ) ) ) ).

% effect_assertI
tff(fact_606_effect__assertE,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_assert(A,P2,X),H,H5,R3,N)
     => ~ ( pp(aa(A,bool,P2,X))
         => ( ( R3 = X )
           => ( ( H5 = H )
             => ( N != one_one(nat) ) ) ) ) ) ).

% effect_assertE
tff(fact_607_set__to__map__empty__iff_I2_J,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( ( aTP_Lamp_bp(A,option(B)) = set_to_map(A,B,S) )
    <=> ( S = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(2)
tff(fact_608_dflt__None__set__def,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( ( S = bot_bot(set(A)) )
       => ( dflt_None_set(A,S) = none(set(A)) ) )
      & ( ( S != bot_bot(set(A)) )
       => ( dflt_None_set(A,S) = aa(set(A),option(set(A)),some(set(A)),S) ) ) ) ).

% dflt_None_set_def
tff(fact_609_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool)] : rel_of(A,B,aTP_Lamp_bp(A,option(B)),P2) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_610_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_611_tap__def,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F3) = heap_Time_Heap2(A,aTP_Lamp_bq(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F3)) ).

% tap_def
tff(fact_612_execute__tap,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_tap(A,F3)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,F3,H)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ).

% execute_tap
tff(fact_613_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).

% add_mono1
tff(fact_614_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))) ) ).

% less_add_one
tff(fact_615_bind__return,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A)] : heap_Time_bind(A,A,F3,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_bd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F3)) ).

% bind_return
tff(fact_616_return__bind,axiom,
    ! [A: $tType,B: $tType,X: B,F3: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),F3) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_br(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),X),F3)) ).

% return_bind
tff(fact_617_lookup__chain,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(B)
     => ! [R3: ref(B),F3: heap_Time_Heap(A)] : heap_Time_bind(B,A,ref_lookup(B,R3),aTP_Lamp_bs(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),F3)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_bd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F3)) ) ).

% lookup_chain
tff(fact_618_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_619_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X),Y)) ).

% bot2E
tff(fact_620_fold__if__return,axiom,
    ! [A: $tType,B2: bool,C3: A,D3: A] :
      ( ( pp(B2)
       => ( aa(A,heap_Time_Heap(A),heap_Time_return(A),C3) = aa(A,heap_Time_Heap(A),heap_Time_return(A),if(A,B2,C3,D3)) ) )
      & ( ~ pp(B2)
       => ( aa(A,heap_Time_Heap(A),heap_Time_return(A),D3) = aa(A,heap_Time_Heap(A),heap_Time_return(A),if(A,B2,C3,D3)) ) ) ) ).

% fold_if_return
tff(fact_621_success__lookupI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),H: heap_ext(product_unit)] : heap_Time_success(A,ref_lookup(A,R3),H) ) ).

% success_lookupI
tff(fact_622_return__wp__rule,axiom,
    ! [A: $tType,Q2: fun(A,assn),X: A] : hoare_hoare_triple(A,aa(A,assn,Q2,X),aa(A,heap_Time_Heap(A),heap_Time_return(A),X),Q2) ).

% return_wp_rule
tff(fact_623_success__returnI,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H) ).

% success_returnI
tff(fact_624_wait__rule,axiom,
    ! [N: nat] : hoare_hoare_triple(product_unit,one_one(assn),heap_Time_wait(N),aTP_Lamp_bt(product_unit,assn)) ).

% wait_rule
tff(fact_625_success__tapI,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_tap(A,F3),H) ).

% success_tapI
tff(fact_626_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_627_bounded__Max__nat,axiom,
    ! [P2: fun(nat,bool),X: nat,M6: nat] :
      ( pp(aa(nat,bool,P2,X))
     => ( ! [X3: nat] :
            ( pp(aa(nat,bool,P2,X3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M6)) )
       => ~ ! [M5: nat] :
              ( pp(aa(nat,bool,P2,M5))
             => ~ ! [X5: nat] :
                    ( pp(aa(nat,bool,P2,X5))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_628_effect__returnE,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H,H5,R3,N)
     => ~ ( ( R3 = X )
         => ( ( H5 = H )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_returnE
tff(fact_629_effect__returnI,axiom,
    ! [A: $tType,H: heap_ext(product_unit),H5: heap_ext(product_unit),X: A] :
      ( ( H = H5 )
     => heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H,H5,X,one_one(nat)) ) ).

% effect_returnI
tff(fact_630_mod__emp__simp,axiom,
    ! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(one_one(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ).

% mod_emp_simp
tff(fact_631_effect__tapE,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_tap(A,F3),H,H5,R3,N)
     => ~ ( ( H5 = H )
         => ( ( R3 = aa(heap_ext(product_unit),A,F3,H) )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_tapE
tff(fact_632_effect__tapI,axiom,
    ! [A: $tType,H5: heap_ext(product_unit),H: heap_ext(product_unit),R3: A,F3: fun(heap_ext(product_unit),A)] :
      ( ( H5 = H )
     => ( ( R3 = aa(heap_ext(product_unit),A,F3,H) )
       => heap_Time_effect(A,heap_Time_tap(A,F3),H,H5,R3,one_one(nat)) ) ) ).

% effect_tapI
tff(fact_633_return__def,axiom,
    ! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),X) = heap_Time_heap(A,aTP_Lamp_bu(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% return_def
tff(fact_634_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P2: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))))] :
      ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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)))))
     => ( ! [F: fun(nat,fun(A,A)),A5: nat,B4: nat,Acc: A] :
            ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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)),A5),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc)))))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B4),A5))
               => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P2,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F,A5),Acc))) )
             => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P2,F),A5),B4),Acc)) ) )
       => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P2,A0),A1),A22),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_635_bind__lift,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(B),G3: fun(B,A)] : heap_Time_bind(B,A,F3,heap_Time_lift(B,A,G3)) = heap_Time_bind(B,A,F3,aTP_Lamp_bv(fun(B,A),fun(B,heap_Time_Heap(A)),G3)) ).

% bind_lift
tff(fact_636_lift__collapse,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: B] : aa(B,heap_Time_Heap(A),heap_Time_lift(B,A,F3),X) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,F3,X)) ).

% lift_collapse
tff(fact_637_execute__lookup,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,ref_lookup(A,R3)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),ref_get(A,H,R3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ).

% execute_lookup
tff(fact_638_execute__return,axiom,
    ! [A: $tType,X: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_bu(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_return
tff(fact_639_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).

% less_numeral_extra(4)
tff(fact_640_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).

% le_numeral_extra(4)
tff(fact_641_change__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [F3: fun(A,A),R3: ref(A)] : ref_change(A,F3,R3) = heap_Time_bind(A,A,ref_lookup(A,R3),aa(ref(A),fun(A,heap_Time_Heap(A)),aTP_Lamp_bx(fun(A,A),fun(ref(A),fun(A,heap_Time_Heap(A))),F3),R3)) ) ).

% change_def
tff(fact_642_map__to__set__empty,axiom,
    ! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_bp(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% map_to_set_empty
tff(fact_643_type__copy__map__cong0,axiom,
    ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M6: fun(B,A),G3: fun(C,B),X: C,N7: fun(D,A),H: fun(C,D),F3: fun(A,E)] :
      ( ( aa(B,A,M6,aa(C,B,G3,X)) = aa(D,A,N7,aa(C,D,H,X)) )
     => ( aa(C,E,aa(fun(C,B),fun(C,E),comp(B,E,C,aa(fun(B,A),fun(B,E),comp(A,E,B,F3),M6)),G3),X) = aa(C,E,aa(fun(C,D),fun(C,E),comp(D,E,C,aa(fun(D,A),fun(D,E),comp(A,E,D,F3),N7)),H),X) ) ) ).

% type_copy_map_cong0
tff(fact_644_comp__cong__left,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: fun(A,B),Y: fun(A,B),F3: fun(C,A)] :
      ( ( X = Y )
     => ( aa(fun(C,A),fun(C,B),comp(A,B,C,X),F3) = aa(fun(C,A),fun(C,B),comp(A,B,C,Y),F3) ) ) ).

% comp_cong_left
tff(fact_645_comp__cong__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,B),Y: fun(A,B),F3: fun(B,C)] :
      ( ( X = Y )
     => ( aa(fun(A,B),fun(A,C),comp(B,C,A,F3),X) = aa(fun(A,B),fun(A,C),comp(B,C,A,F3),Y) ) ) ).

% comp_cong_right
tff(fact_646_fun__comp__eq__conv,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(C,B),G3: fun(A,C),Fg: fun(A,B)] :
      ( ( aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3) = Fg )
    <=> ! [X4: A] : aa(C,B,F3,aa(A,C,G3,X4)) = aa(A,B,Fg,X4) ) ).

% fun_comp_eq_conv
tff(fact_647_comp__cong,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F3: fun(B,A),G3: fun(C,B),X: C,F5: fun(D,A),G4: fun(E,D),X8: E] :
      ( ( aa(B,A,F3,aa(C,B,G3,X)) = aa(D,A,F5,aa(E,D,G4,X8)) )
     => ( aa(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3),X) = aa(E,A,aa(fun(E,D),fun(E,A),comp(D,A,E,F5),G4),X8) ) ) ).

% comp_cong
tff(fact_648_execute__heap,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_execute(A,heap_Time_heap(A,F3)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F3) ).

% execute_heap
tff(fact_649_heap__def,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_heap(A,F3) = heap_Time_Heap2(A,aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F3)) ).

% heap_def
tff(fact_650_map__to__set__inverse,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : set_to_map(A,B,map_to_set(A,B,M)) = M ).

% map_to_set_inverse
tff(fact_651_success__heapI,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_heap(A,F3),H) ).

% success_heapI
tff(fact_652_success__changeI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [F3: fun(A,A),R3: ref(A),H: heap_ext(product_unit)] : heap_Time_success(A,ref_change(A,F3,R3),H) ) ).

% success_changeI
tff(fact_653_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% is_num_normalize(1)
tff(fact_654_map__to__set__empty__iff_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M) )
    <=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).

% map_to_set_empty_iff(2)
tff(fact_655_map__to__set__empty__iff_I1_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( map_to_set(A,B,M) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).

% map_to_set_empty_iff(1)
tff(fact_656_update__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),Y: A,X: A] : hoare_hoare_triple(product_unit,sngr_assn(A,P3,Y),ref_update(A,P3,X),aa(A,fun(product_unit,assn),aTP_Lamp_by(ref(A),fun(A,fun(product_unit,assn)),P3),X)) ) ).

% update_rule
tff(fact_657_lookup__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A)] : ref_lookup(A,R3) = heap_Time_tap(A,aTP_Lamp_bz(ref(A),fun(heap_ext(product_unit),A),R3)) ) ).

% lookup_def
tff(fact_658_effect__lookupE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),X: A,N: nat] :
          ( heap_Time_effect(A,ref_lookup(A,R3),H,H5,X,N)
         => ~ ( ( H5 = H )
             => ( ( X = ref_get(A,H,R3) )
               => ( N != one_one(nat) ) ) ) ) ) ).

% effect_lookupE
tff(fact_659_effect__lookupI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H5: heap_ext(product_unit),H: heap_ext(product_unit),X: A,R3: ref(A),N: nat] :
          ( ( H5 = H )
         => ( ( X = ref_get(A,H,R3) )
           => ( ( N = one_one(nat) )
             => heap_Time_effect(A,ref_lookup(A,R3),H,H5,X,N) ) ) ) ) ).

% effect_lookupI
tff(fact_660_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
      ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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))),F3),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)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2)))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = Acc2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A3),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_661_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb2: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb2,Xc) = Y )
     => ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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))),X),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)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb2),Xc)))))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
               => ( Y = Xc ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
               => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb2,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) )
           => ~ pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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))),X),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)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb2),Xc))))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_662_ref__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: A] : hoare_hoare_triple(ref(A),one_one(assn),ref_ref(A,X),aa(A,fun(ref(A),assn),aTP_Lamp_ca(A,fun(ref(A),assn)),X)) ) ).

% ref_rule
tff(fact_663_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_cb(list(A),fun(array(A),assn)),Xs)) ) ).

% of_list_rule
tff(fact_664_Set_Ois__empty__def,axiom,
    ! [A: $tType,A6: set(A)] :
      ( is_empty(A,A6)
    <=> ( A6 = bot_bot(set(A)) ) ) ).

% Set.is_empty_def
tff(fact_665_Heap__lub__empty,axiom,
    ! [A: $tType] : heap_Time_Heap_lub(A,bot_bot(set(heap_Time_Heap(A)))) = heap_Time_Heap2(A,aTP_Lamp_az(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).

% Heap_lub_empty
tff(fact_666_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A3: nat,F3: fun(nat,fun(A,A)),Acc2: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = Acc2 ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A3),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_667_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb2: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb2,Xc) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa2))
         => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb2,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_668_mod__h__bot__iff_I1_J,axiom,
    ! [B2: bool,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(pure_assn(B2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> pp(B2) ) ).

% mod_h_bot_iff(1)
tff(fact_669_fun_Omap__ident,axiom,
    ! [A: $tType,D: $tType,T5: fun(D,A)] : aa(fun(D,A),fun(D,A),comp(A,A,D,aTP_Lamp_cc(A,A)),T5) = T5 ).

% fun.map_ident
tff(fact_670_pure__assn__eq__conv,axiom,
    ! [P2: bool,Q2: bool] :
      ( ( pure_assn(P2) = pure_assn(Q2) )
    <=> ( pp(P2)
      <=> pp(Q2) ) ) ).

% pure_assn_eq_conv
tff(fact_671_merge__pure__or,axiom,
    ! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fdisj(A3,B2)) ).

% merge_pure_or
tff(fact_672_pure__assn__eq__emp__iff,axiom,
    ! [P2: bool] :
      ( ( pure_assn(P2) = one_one(assn) )
    <=> pp(P2) ) ).

% pure_assn_eq_emp_iff
tff(fact_673_pure__true,axiom,
    pure_assn(fTrue) = one_one(assn) ).

% pure_true
tff(fact_674_pure__assn__eq__false__iff,axiom,
    ! [P2: bool] :
      ( ( pure_assn(P2) = bot_bot(assn) )
    <=> ~ pp(P2) ) ).

% pure_assn_eq_false_iff
tff(fact_675_pure__false,axiom,
    pure_assn(fFalse) = bot_bot(assn) ).

% pure_false
tff(fact_676_norm__pre__pure__iff__sng,axiom,
    ! [A: $tType,B2: bool,F3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( hoare_hoare_triple(A,pure_assn(B2),F3,Q2)
    <=> ( pp(B2)
       => hoare_hoare_triple(A,one_one(assn),F3,Q2) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_677_norm__pre__pure__rule2,axiom,
    ! [A: $tType,B2: bool,F3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( ( pp(B2)
       => hoare_hoare_triple(A,one_one(assn),F3,Q2) )
     => hoare_hoare_triple(A,pure_assn(B2),F3,Q2) ) ).

% norm_pre_pure_rule2
tff(fact_678_lift__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : heap_Time_lift(A,B,F3) = aa(fun(A,B),fun(A,heap_Time_Heap(B)),comp(B,heap_Time_Heap(B),A,heap_Time_return(B)),F3) ).

% lift_def
tff(fact_679_lookup__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),X: A] : hoare_hoare_triple(A,sngr_assn(A,P3,X),ref_lookup(A,P3),aa(A,fun(A,assn),aTP_Lamp_cd(ref(A),fun(A,fun(A,assn)),P3),X)) ) ).

% lookup_rule
tff(fact_680_K__record__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,C3: B,F3: fun(A,C),X5: A] : aa(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,aTP_Lamp_ce(B,fun(C,B),C3)),F3),X5) = C3 ).

% K_record_comp
tff(fact_681_execute__ureturn,axiom,
    ! [A: $tType,X: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_cf(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_ureturn
tff(fact_682_mlex__leq,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2))) ) ) ).

% mlex_leq
tff(fact_683_mlex__iff,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
        | ( ( aa(A,nat,F3,X) = aa(A,nat,F3,Y) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ).

% mlex_iff
tff(fact_684_mlex__less,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2))) ) ).

% mlex_less
tff(fact_685_ent__pure__post__iff__sng,axiom,
    ! [P2: assn,B2: bool] :
      ( entails(P2,pure_assn(B2))
    <=> ( ! [H7: product_prod(heap_ext(product_unit),set(nat))] :
            ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H7))
           => pp(B2) )
        & entails(P2,one_one(assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_686_pure__assn__raw_Osimps,axiom,
    ! [B: $tType,A: $tType,B2: bool,H: A,As: set(B)] :
      ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,B2),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As)))
    <=> ( ( As = bot_bot(set(B)) )
        & pp(B2) ) ) ).

% pure_assn_raw.simps
tff(fact_687_pure__assn__raw_Oelims_I1_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B)),Y: bool] :
      ( ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
      <=> pp(Y) )
     => ~ ! [H2: A,As2: set(B)] :
            ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2) )
           => ( pp(Y)
            <=> ~ ( ( As2 = bot_bot(set(B)) )
                  & pp(X) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_688_pure__assn__raw_Oelims_I2_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ~ ! [H2: A,As2: set(B)] :
            ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2) )
           => ~ ( ( As2 = bot_bot(set(B)) )
                & pp(X) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_689_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_690_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_691_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add_0
tff(fact_692_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_693_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_694_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_695_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_696_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_697_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_698_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_699_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.right_neutral
tff(fact_700_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_701_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% neq0_conv
tff(fact_702_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A3)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_703_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% le0
tff(fact_704_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A3)) ).

% bot_nat_0.extremum
tff(fact_705_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_706_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_707_merge__pure__star,axiom,
    ! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fconj(A3,B2)) ).

% merge_pure_star
tff(fact_708_timeFrame__zero,axiom,
    ! [A: $tType,H: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,zero_zero(nat),H) = H ).

% timeFrame_zero
tff(fact_709_star__false__left,axiom,
    ! [P2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P2) = bot_bot(assn) ).

% star_false_left
tff(fact_710_star__false__right,axiom,
    ! [P2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),bot_bot(assn)) = bot_bot(assn) ).

% star_false_right
tff(fact_711_triv__exI,axiom,
    ! [A: $tType,Q2: fun(A,assn),X: A] : entails(aa(A,assn,Q2,X),ex_assn(A,Q2)) ).

% triv_exI
tff(fact_712_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_713_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_714_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_715_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_716_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_717_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_718_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_719_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_720_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_721_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_722_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_723_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_724_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% add_gr_0
tff(fact_725_less__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_726_mod__pure__star__dist,axiom,
    ! [P2: assn,B2: bool,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),pure_assn(B2))),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H))
        & pp(B2) ) ) ).

% mod_pure_star_dist
tff(fact_727_ent__pure__pre__iff,axiom,
    ! [P2: assn,B2: bool,Q2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),pure_assn(B2)),Q2)
    <=> ( pp(B2)
       => entails(P2,Q2) ) ) ).

% ent_pure_pre_iff
tff(fact_728_norm__pre__pure__iff,axiom,
    ! [A: $tType,P2: assn,B2: bool,F3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),pure_assn(B2)),F3,Q2)
    <=> ( pp(B2)
       => hoare_hoare_triple(A,P2,F3,Q2) ) ) ).

% norm_pre_pure_iff
tff(fact_729_ent__false__iff,axiom,
    ! [P2: assn] :
      ( entails(P2,bot_bot(assn))
    <=> ! [H7: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H7)) ) ).

% ent_false_iff
tff(fact_730_ent__pure__pre__iff__sng,axiom,
    ! [B2: bool,Q2: assn] :
      ( entails(pure_assn(B2),Q2)
    <=> ( pp(B2)
       => entails(one_one(assn),Q2) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_731_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),X: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,P3,X)),snga_assn(A,P3,Y)) = bot_bot(assn) ) ).

% snga_same_false
tff(fact_732_sngr__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),X: A,Y: A] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P3,X)),sngr_assn(A,P3,Y)) = bot_bot(assn) ) ).

% sngr_same_false
tff(fact_733_mod__h__bot__iff_I5_J,axiom,
    ! [P2: assn,Q2: assn,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_iff(5)
tff(fact_734_ent__pure__post__iff,axiom,
    ! [P2: assn,Q2: assn,B2: bool] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),pure_assn(B2)))
    <=> ( ! [H7: product_prod(heap_ext(product_unit),set(nat))] :
            ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H7))
           => pp(B2) )
        & entails(P2,Q2) ) ) ).

% ent_pure_post_iff
tff(fact_735_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_736_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_cg(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_737_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_mono
tff(fact_738_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_mono'
tff(fact_739_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3))) ) ).

% zero_le_square
tff(fact_740_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ).

% split_mult_pos_le
tff(fact_741_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_left_mono_neg
tff(fact_742_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_743_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_left_mono
tff(fact_744_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_right_mono_neg
tff(fact_745_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_right_mono
tff(fact_746_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_le_0_iff
tff(fact_747_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ).

% split_mult_neg_le
tff(fact_748_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_749_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_750_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_751_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_752_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_753_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_754_ent__iffI,axiom,
    ! [A6: assn,B5: assn] :
      ( entails(A6,B5)
     => ( entails(B5,A6)
       => ( A6 = B5 ) ) ) ).

% ent_iffI
tff(fact_755_ent__refl,axiom,
    ! [P2: assn] : entails(P2,P2) ).

% ent_refl
tff(fact_756_ent__trans,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( entails(P2,Q2)
     => ( entails(Q2,R2)
       => entails(P2,R2) ) ) ).

% ent_trans
tff(fact_757_ent__star__mono,axiom,
    ! [P2: assn,P4: assn,Q2: assn,Q3: assn] :
      ( entails(P2,P4)
     => ( entails(Q2,Q3)
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P4),Q3)) ) ) ).

% ent_star_mono
tff(fact_758_assn__times__comm,axiom,
    ! [P2: assn,Q2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),P2) ).

% assn_times_comm
tff(fact_759_assn__times__assoc,axiom,
    ! [P2: assn,Q2: assn,R2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2)),R2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),R2)) ).

% assn_times_assoc
tff(fact_760_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_761_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A))) ) ).

% not_square_less_zero
tff(fact_762_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_less_0_iff
tff(fact_763_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_764_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_765_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_766_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_767_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_768_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_769_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_770_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_771_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_772_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_773_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_strict_left_mono
tff(fact_774_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_775_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_776_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_strict_right_mono
tff(fact_777_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_778_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_779_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F3: fun(fun(A,B),C),G3: C] :
          ( ! [X3: fun(A,B)] : aa(fun(A,B),C,F3,X3) = G3
         => ( aa(fun(A,B),C,F3,aTP_Lamp_ch(A,B)) = G3 ) ) ) ).

% fun_cong_unused_0
tff(fact_780_effect__ureturnI,axiom,
    ! [A: $tType,H: heap_ext(product_unit),H5: heap_ext(product_unit),X: A] :
      ( ( H = H5 )
     => heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H,H5,X,zero_zero(nat)) ) ).

% effect_ureturnI
tff(fact_781_effect__ureturnE,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H,H5,R3,N)
     => ~ ( ( R3 = X )
         => ( ( H5 = H )
           => ( N != zero_zero(nat) ) ) ) ) ).

% effect_ureturnE
tff(fact_782_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_783_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_784_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_785_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_786_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_787_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_788_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_789_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_790_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_791_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_792_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_793_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_794_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_795_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_796_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).

% sum_squares_ge_zero
tff(fact_797_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),A3)) ) ) ) ).

% mult_left_le
tff(fact_798_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ) ).

% mult_le_one
tff(fact_799_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_800_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_801_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_802_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E3)),C3) ) ).

% combine_common_factor
tff(fact_803_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% distrib_right
tff(fact_804_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).

% distrib_left
tff(fact_805_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% comm_semiring_class.distrib
tff(fact_806_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).

% ring_class.ring_distribs(1)
tff(fact_807_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ring_class.ring_distribs(2)
tff(fact_808_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),zero_zero(A))) ) ).

% le_numeral_extra(3)
tff(fact_809_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).

% zero_le
tff(fact_810_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),zero_zero(A))) ) ).

% less_numeral_extra(3)
tff(fact_811_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_812_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_813_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).

% not_less_zero
tff(fact_814_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).

% gr_zeroI
tff(fact_815_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add.group_left_neutral
tff(fact_816_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.comm_neutral
tff(fact_817_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% comm_monoid_add_class.add_0
tff(fact_818_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% verit_sum_simplify
tff(fact_819_ent__fwd,axiom,
    ! [P2: assn,H: product_prod(heap_ext(product_unit),set(nat)),Q2: assn] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H))
     => ( entails(P2,Q2)
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),H)) ) ) ).

% ent_fwd
tff(fact_820_entailsD,axiom,
    ! [P2: assn,Q2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( entails(P2,Q2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H))
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),H)) ) ) ).

% entailsD
tff(fact_821_entailsI,axiom,
    ! [P2: assn,Q2: assn] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H2))
         => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),H2)) )
     => entails(P2,Q2) ) ).

% entailsI
tff(fact_822_entails__def,axiom,
    ! [P2: assn,Q2: assn] :
      ( entails(P2,Q2)
    <=> ! [H7: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H7))
         => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),H7)) ) ) ).

% entails_def
tff(fact_823_mod__starD,axiom,
    ! [A6: assn,B5: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A6),B5)),H))
     => ? [H12: product_prod(heap_ext(product_unit),set(nat)),H23: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(A6),H12))
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(B5),H23)) ) ) ).

% mod_starD
tff(fact_824_mod__starE,axiom,
    ! [A3: assn,B2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B2)),H))
     => ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(A3),X_1))
         => ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(B2),H_2)) ) ) ).

% mod_starE
tff(fact_825_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P2: fun(A,bool),X: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V,X3) = zero_zero(nat) )
         => pp(aa(A,bool,P2,X3)) )
     => ( ! [X3: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X3)))
           => ( ~ pp(aa(A,bool,P2,X3))
             => ? [Y5: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y5)),aa(A,nat,V,X3)))
                  & ~ pp(aa(A,bool,P2,Y5)) ) ) )
       => pp(aa(A,bool,P2,X)) ) ) ).

% infinite_descent0_measure
tff(fact_826_infinite__descent0,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
           => ( ~ pp(aa(nat,bool,P2,N4))
             => ? [M4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N4))
                  & ~ pp(aa(nat,bool,P2,M4)) ) ) )
       => pp(aa(nat,bool,P2,N)) ) ) ).

% infinite_descent0
tff(fact_827_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_828_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_zeroE
tff(fact_829_not__less0,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% not_less0
tff(fact_830_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_831_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% gr0I
tff(fact_832_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_833_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_834_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_835_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_836_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% less_eq_nat.simps(1)
tff(fact_837_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_838_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_839_cons__post__rule,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn),Q3: fun(A,assn)] :
      ( hoare_hoare_triple(A,P2,C3,Q2)
     => ( ! [X3: A] : entails(aa(A,assn,Q2,X3),aa(A,assn,Q3,X3))
       => hoare_hoare_triple(A,P2,C3,Q3) ) ) ).

% cons_post_rule
tff(fact_840_cons__rule,axiom,
    ! [A: $tType,P2: assn,P4: assn,Q2: fun(A,assn),Q3: fun(A,assn),C3: heap_Time_Heap(A)] :
      ( entails(P2,P4)
     => ( ! [X3: A] : entails(aa(A,assn,Q2,X3),aa(A,assn,Q3,X3))
       => ( hoare_hoare_triple(A,P4,C3,Q2)
         => hoare_hoare_triple(A,P2,C3,Q3) ) ) ) ).

% cons_rule
tff(fact_841_ent__disjE,axiom,
    ! [A6: assn,C4: assn,B5: assn] :
      ( entails(A6,C4)
     => ( entails(B5,C4)
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A6),B5),C4) ) ) ).

% ent_disjE
tff(fact_842_ent__disjI1,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2),R2)
     => entails(P2,R2) ) ).

% ent_disjI1
tff(fact_843_ent__disjI2,axiom,
    ! [P2: assn,Q2: assn,R2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2),R2)
     => entails(Q2,R2) ) ).

% ent_disjI2
tff(fact_844_ent__disjI1_H,axiom,
    ! [A6: assn,B5: assn,C4: assn] :
      ( entails(A6,B5)
     => entails(A6,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B5),C4)) ) ).

% ent_disjI1'
tff(fact_845_ent__disjI2_H,axiom,
    ! [A6: assn,C4: assn,B5: assn] :
      ( entails(A6,C4)
     => entails(A6,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B5),C4)) ) ).

% ent_disjI2'
tff(fact_846_ent__disjI1__direct,axiom,
    ! [A6: assn,B5: assn] : entails(A6,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A6),B5)) ).

% ent_disjI1_direct
tff(fact_847_ent__disjI2__direct,axiom,
    ! [B5: assn,A6: assn] : entails(B5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A6),B5)) ).

% ent_disjI2_direct
tff(fact_848_star__or__dist1,axiom,
    ! [A6: assn,B5: assn,C4: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A6),B5)),C4) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A6),C4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B5),C4)) ).

% star_or_dist1
tff(fact_849_star__or__dist2,axiom,
    ! [C4: assn,A6: assn,B5: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A6),B5)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),A6)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),B5)) ).

% star_or_dist2
tff(fact_850_assn__one__left,axiom,
    ! [P2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),P2) = P2 ).

% assn_one_left
tff(fact_851_ent__false,axiom,
    ! [P2: assn] : entails(bot_bot(assn),P2) ).

% ent_false
tff(fact_852_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_853_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_854_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_855_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_856_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_857_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_858_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_859_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_860_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A3: A,Y: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_861_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ci(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_862_cons__pre__rule,axiom,
    ! [A: $tType,P2: assn,P4: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( entails(P2,P4)
     => ( hoare_hoare_triple(A,P4,C3,Q2)
       => hoare_hoare_triple(A,P2,C3,Q2) ) ) ).

% cons_pre_rule
tff(fact_863_frame__rule,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn),R2: assn] :
      ( hoare_hoare_triple(A,P2,C3,Q2)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),R2),C3,aa(assn,fun(A,assn),aTP_Lamp_cj(fun(A,assn),fun(assn,fun(A,assn)),Q2),R2)) ) ).

% frame_rule
tff(fact_864_ent__ex__postI,axiom,
    ! [A: $tType,P2: assn,Q2: fun(A,assn),X: A] :
      ( entails(P2,aa(A,assn,Q2,X))
     => entails(P2,ex_assn(A,Q2)) ) ).

% ent_ex_postI
tff(fact_865_ent__ex__preI,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] :
      ( ! [X3: A] : entails(aa(A,assn,P2,X3),Q2)
     => entails(ex_assn(A,P2),Q2) ) ).

% ent_ex_preI
tff(fact_866_ex__distrib__star,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_cj(fun(A,assn),fun(assn,fun(A,assn)),P2),Q2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,P2)),Q2) ).

% ex_distrib_star
tff(fact_867_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A3: A,Y: A,U: A,V2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V2))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_868_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N))) ) ) ) ).

% less_1_mult
tff(fact_869_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2)) ) ) ) ).

% add_decreasing
tff(fact_870_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_increasing
tff(fact_871_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_872_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_increasing2
tff(fact_873_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_nonneg_nonneg
tff(fact_874_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_nonpos
tff(fact_875_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_876_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_877_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).

% not_one_le_zero
tff(fact_878_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_879_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one_class.zero_le_one
tff(fact_880_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% pos_add_strict
tff(fact_881_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ ! [C2: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
               => ( C2 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_882_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_pos
tff(fact_883_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_884_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_885_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% less_numeral_extra(1)
tff(fact_886_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).

% not_one_less_zero
tff(fact_887_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one
tff(fact_888_ex__least__nat__le,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
       => ? [K: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
               => ~ pp(aa(nat,bool,P2,I4)) )
            & pp(aa(nat,bool,P2,K)) ) ) ) ).

% ex_least_nat_le
tff(fact_889_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ? [K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K) = J2 ) ) ) ).

% less_imp_add_positive
tff(fact_890_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),X))
    <=> ( X != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_891_norm__pre__pure__rule1,axiom,
    ! [A: $tType,B2: bool,P2: assn,F3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( ( pp(B2)
       => hoare_hoare_triple(A,P2,F3,Q2) )
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),pure_assn(B2)),F3,Q2) ) ).

% norm_pre_pure_rule1
tff(fact_892_bind__ureturn,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A)] : heap_Time_bind(A,A,F3,heap_Time_ureturn(A)) = F3 ).

% bind_ureturn
tff(fact_893_ureturn__bind,axiom,
    ! [A: $tType,B: $tType,X: B,F3: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X),F3) = aa(B,heap_Time_Heap(A),F3,X) ).

% ureturn_bind
tff(fact_894_success__ureturnI,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H) ).

% success_ureturnI
tff(fact_895_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ck(nat,fun(nat,bool)),aTP_Lamp_cl(nat,fun(nat,bool)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_896_ureturn__def,axiom,
    ! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X) = heap_Time_heap(A,aTP_Lamp_cf(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% ureturn_def
tff(fact_897_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_898_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_899_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_900_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_901_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_strict_increasing
tff(fact_902_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_strict_increasing2
tff(fact_903_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),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_904_return__sp__rule,axiom,
    ! [A: $tType,P2: assn,X: A] : hoare_hoare_triple(A,P2,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),aa(A,fun(A,assn),aTP_Lamp_cm(assn,fun(A,fun(A,assn)),P2),X)) ).

% return_sp_rule
tff(fact_905_pure__assn__raw_Oelims_I3_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( ~ pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ~ ! [H2: A,As2: set(B)] :
            ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2) )
           => ( ( As2 = bot_bot(set(B)) )
              & pp(X) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_906_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),one_one(A)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_mult_one_interval
tff(fact_907_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_908_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_909_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_910_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z4)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z4)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_911_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_912_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_epsilon
tff(fact_913_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q5: A,R3: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),R3))
        <=> ( R3 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_914_pure__assn__raw_Opelims_I3_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( ~ pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa2)))
       => ~ ! [H2: A,As2: set(B)] :
              ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2) )
             => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2))))
               => ( ( As2 = bot_bot(set(B)) )
                  & pp(X) ) ) ) ) ) ).

% pure_assn_raw.pelims(3)
tff(fact_915_pure__assn__raw_Opelims_I2_J,axiom,
    ! [B: $tType,A: $tType,X: bool,Xa2: product_prod(A,set(B))] :
      ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
     => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa2)))
       => ~ ! [H2: A,As2: set(B)] :
              ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2) )
             => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2))))
               => ~ ( ( As2 = bot_bot(set(B)) )
                    & pp(X) ) ) ) ) ) ).

% pure_assn_raw.pelims(2)
tff(fact_916_mult__cancel2,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2) )
    <=> ( ( M = N )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_917_mult__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N) )
    <=> ( ( M = N )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_918_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_919_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_920_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_921_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_922_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_923_mult__less__cancel2,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% mult_less_cancel2
tff(fact_924_mult__le__cancel2,axiom,
    ! [M: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% mult_le_cancel2
tff(fact_925_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_926_le__cube,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_927_le__square,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_square
tff(fact_928_mult__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),L))) ) ) ).

% mult_le_mono
tff(fact_929_mult__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K2))) ) ).

% mult_le_mono1
tff(fact_930_mult__le__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J2))) ) ).

% mult_le_mono2
tff(fact_931_add__mult__distrib2,axiom,
    ! [K2: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) ).

% add_mult_distrib2
tff(fact_932_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)) ).

% add_mult_distrib
tff(fact_933_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_934_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_935_mult__less__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J2))) ) ) ).

% mult_less_mono2
tff(fact_936_mult__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K2))) ) ) ).

% mult_less_mono1
tff(fact_937_mlex__snd__decrI,axiom,
    ! [A3: nat,A4: nat,B2: nat,B3: nat,N7: nat] :
      ( ( A3 = A4 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N7)),B3))) ) ) ).

% mlex_snd_decrI
tff(fact_938_mlex__fst__decrI,axiom,
    ! [A3: nat,A4: nat,B2: nat,N7: nat,B3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N7))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B3),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N7)),B3))) ) ) ) ).

% mlex_fst_decrI
tff(fact_939_mlex__bound,axiom,
    ! [A3: nat,A6: nat,B2: nat,N7: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N7))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N7))) ) ) ).

% mlex_bound
tff(fact_940_mlex__leI,axiom,
    ! [A3: nat,A4: nat,B2: nat,B3: nat,N7: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),A4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N7)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N7)),B3))) ) ) ).

% mlex_leI
tff(fact_941_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_942_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X5)) ) ).

% linordered_field_no_lb
tff(fact_943_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X5: A] :
        ? [X_1: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),X_1)) ) ).

% linordered_field_no_ub
tff(fact_944_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z4)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z4)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mult_less_iff1
tff(fact_945_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: 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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_946_pure__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType,B: $tType,X: bool,Xa2: product_prod(A,set(B)),Y: bool] :
      ( ( pp(aa(product_prod(A,set(B)),bool,pure_assn_raw(A,B,X),Xa2))
      <=> pp(Y) )
     => ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),Xa2)))
       => ~ ! [H2: A,As2: set(B)] :
              ( ( Xa2 = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2) )
             => ( ( pp(Y)
                <=> ( ( As2 = bot_bot(set(B)) )
                    & pp(X) ) )
               => ~ pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As2)))) ) ) ) ) ).

% pure_assn_raw.pelims(1)
tff(fact_947_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_948_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_949_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_950_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R3: A,A3: A,B2: A,C3: A,D3: A] :
          ( ( R3 != zero_zero(A) )
         => ( ( ( A3 = B2 )
              & ( C3 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R3),C3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_951_freeze__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Xs: list(A)] : hoare_hoare_triple(list(A),snga_assn(A,A3,Xs),array_freeze(A,A3),aa(list(A),fun(list(A),assn),aTP_Lamp_cn(array(A),fun(list(A),fun(list(A),assn)),A3),Xs)) ) ).

% freeze_rule
tff(fact_952_precise__extr__pure_I2_J,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn)),P2: bool] :
      ( precise(A,B,aa(bool,fun(A,fun(B,assn)),aTP_Lamp_co(fun(A,fun(B,assn)),fun(bool,fun(A,fun(B,assn))),R2),P2))
    <=> ( pp(P2)
       => precise(A,B,R2) ) ) ).

% precise_extr_pure(2)
tff(fact_953_precise__extr__pure_I1_J,axiom,
    ! [B: $tType,A: $tType,P2: bool,R2: fun(A,fun(B,assn))] :
      ( precise(A,B,aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_cp(bool,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),P2),R2))
    <=> ( pp(P2)
       => precise(A,B,R2) ) ) ).

% precise_extr_pure(1)
tff(fact_954_Euclid__induct,axiom,
    ! [P2: fun(nat,fun(nat,bool)),A3: nat,B2: nat] :
      ( ! [A5: nat,B4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),B4))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,B4),A5)) )
     => ( ! [A5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),zero_zero(nat)))
       => ( ! [A5: nat,B4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),B4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B4))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A3),B2)) ) ) ) ).

% Euclid_induct
tff(fact_955_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_956_snga__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(list(A),array(A),aTP_Lamp_cb(list(A),fun(array(A),assn))) ) ).

% snga_prec
tff(fact_957_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J2: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),K2)) = 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),I2),J2)),U)),K2) ).

% left_add_mult_distrib
tff(fact_958_sngr__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(A,ref(A),aTP_Lamp_ca(A,fun(ref(A),assn))) ) ).

% sngr_prec
tff(fact_959_preciseD_H,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn)),A3: A,P3: B,F4: assn,H: product_prod(heap_ext(product_unit),set(nat)),A4: A,F6: assn] :
      ( precise(A,B,R2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A3),P3)),F4)),H))
       => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A4),P3)),F6)),H))
         => ( A3 = A4 ) ) ) ) ).

% preciseD'
tff(fact_960_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_961_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( ( A3 != B2 )
            & ( C3 != D3 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% crossproduct_noteq
tff(fact_962_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W2: A,Y: A,X: A,Z4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Z4)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
        <=> ( ( W2 = X )
            | ( Y = Z4 ) ) ) ) ).

% crossproduct_eq
tff(fact_963_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_964_ent__mp,axiom,
    ! [P2: assn,Q2: assn] : entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),wand_assn(P2,Q2)),Q2) ).

% ent_mp
tff(fact_965_ent__wandI,axiom,
    ! [Q2: assn,P2: assn,R2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),P2),R2)
     => entails(P2,wand_assn(Q2,R2)) ) ).

% ent_wandI
tff(fact_966_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,B2),M)),aa(nat,A,power_power(A,B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_967_mod__star__conv,axiom,
    ! [A6: assn,B5: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A6),B5)),H))
    <=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: set(nat)] :
          ( ( H = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22)) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(A6),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As1)))
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(B5),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As22))) ) ) ).

% mod_star_conv
tff(fact_968_star__assnI,axiom,
    ! [P2: assn,H: heap_ext(product_unit),As: set(nat),Q2: assn,As5: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As5)))
       => ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As5) = bot_bot(set(nat)) )
         => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5)))) ) ) ) ).

% star_assnI
tff(fact_969_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_970_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,B2),X)),aa(nat,A,power_power(A,B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ) ) ).

% power_increasing_iff
tff(fact_971_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_972_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_973_Int__iff,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))
    <=> ( pp(aa(set(A),bool,member(A,C3),A6))
        & pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% Int_iff
tff(fact_974_IntI,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),A6))
     => ( pp(aa(set(A),bool,member(A,C3),B5))
       => pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5))) ) ) ).

% IntI
tff(fact_975_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z4)) ) ) ) ).

% le_inf_iff
tff(fact_976_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% inf.bounded_iff
tff(fact_977_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,power_power(nat,X),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_978_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = X ) ).

% inf_sup_absorb
tff(fact_979_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = X ) ).

% sup_inf_absorb
tff(fact_980_Int__subset__iff,axiom,
    ! [A: $tType,C4: set(A),A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A6))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B5)) ) ) ).

% Int_subset_iff
tff(fact_981_inf__Some,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: 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),X)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) ) ).

% inf_Some
tff(fact_982_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T8: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T8),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T8)) = T8 ).

% Int_Un_eq(4)
tff(fact_983_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T8: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T8)) = S ).

% Int_Un_eq(3)
tff(fact_984_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T8: 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)),S),T8)),T8) = T8 ).

% Int_Un_eq(2)
tff(fact_985_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T8: 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)),S),T8)),S) = S ).

% Int_Un_eq(1)
tff(fact_986_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T8: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T8),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T8)) = T8 ).

% Un_Int_eq(4)
tff(fact_987_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T8: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T8)) = S ).

% Un_Int_eq(3)
tff(fact_988_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T8: 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)),sup_sup(set(A)),S),T8)),T8) = T8 ).

% Un_Int_eq(2)
tff(fact_989_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T8: 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)),sup_sup(set(A)),S),T8)),S) = S ).

% Un_Int_eq(1)
tff(fact_990_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( ( aa(nat,A,power_power(A,A3),M) = aa(nat,A,power_power(A,A3),N) )
          <=> ( M = N ) ) ) ) ).

% power_inject_exp
tff(fact_991_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_992_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_divide_1_iff
tff(fact_993_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_994_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_995_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_996_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_997_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_998_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_divide_1_iff
tff(fact_999_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,B2),X)),aa(nat,A,power_power(A,B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1000_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( aa(nat,A,power_power(A,A3),N) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_1001_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1002_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1003_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,B2),M)),aa(nat,A,power_power(A,B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1004_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,B2),N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_1005_Int__left__commute,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C4)) ).

% Int_left_commute
tff(fact_1006_Int__left__absorb,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) ).

% Int_left_absorb
tff(fact_1007_Int__commute,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A6) ).

% Int_commute
tff(fact_1008_Int__absorb,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),A6) = A6 ).

% Int_absorb
tff(fact_1009_Int__assoc,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: 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)),A6),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ).

% Int_assoc
tff(fact_1010_IntD2,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))
     => pp(aa(set(A),bool,member(A,C3),B5)) ) ).

% IntD2
tff(fact_1011_IntD1,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))
     => pp(aa(set(A),bool,member(A,C3),A6)) ) ).

% IntD1
tff(fact_1012_IntE,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))
     => ~ ( pp(aa(set(A),bool,member(A,C3),A6))
         => ~ pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% IntE
tff(fact_1013_Int__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_cq(set(A),fun(set(A),fun(A,bool)),A6),B5)) ).

% Int_def
tff(fact_1014_Int__Collect,axiom,
    ! [A: $tType,X: A,A6: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(fun(A,bool),set(A),collect(A),P2))))
    <=> ( pp(aa(set(A),bool,member(A,X),A6))
        & pp(aa(A,bool,P2,X)) ) ) ).

% Int_Collect
tff(fact_1015_Collect__conj__eq,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ah(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,bool),set(A),collect(A),P2)),aa(fun(A,bool),set(A),collect(A),Q2)) ).

% Collect_conj_eq
tff(fact_1016_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_sup_ord(2)
tff(fact_1017_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_sup_ord(1)
tff(fact_1018_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_le1
tff(fact_1019_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_le2
tff(fact_1020_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).

% le_infE
tff(fact_1021_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))) ) ) ) ).

% le_infI
tff(fact_1022_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C3),D3))) ) ) ) ).

% inf_mono
tff(fact_1023_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI1
tff(fact_1024_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI2
tff(fact_1025_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.orderE
tff(fact_1026_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% inf.orderI
tff(fact_1027_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),X3))
         => ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Y3))
           => ( ! [X3: A,Y3: A,Z3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_1028_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_1029_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb1
tff(fact_1030_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_1031_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_1032_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_1033_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% inf.boundedE
tff(fact_1034_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))) ) ) ) ).

% inf.boundedI
tff(fact_1035_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4))) ) ) ) ).

% inf_greatest
tff(fact_1036_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.order_iff
tff(fact_1037_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3)) ) ).

% inf.cobounded1
tff(fact_1038_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2)) ) ).

% inf.cobounded2
tff(fact_1039_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb_iff1
tff(fact_1040_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_1041_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).

% inf.coboundedI1
tff(fact_1042_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).

% inf.coboundedI2
tff(fact_1043_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% less_infI1
tff(fact_1044_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% less_infI2
tff(fact_1045_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb3
tff(fact_1046_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_1047_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% inf.strict_boundedE
tff(fact_1048_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_1049_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).

% inf.strict_coboundedI1
tff(fact_1050_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).

% inf.strict_coboundedI2
tff(fact_1051_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z4: A] :
          ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Z3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z4)) ) ) ) ).

% distrib_imp1
tff(fact_1052_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z4: A] :
          ( ! [X3: A,Y3: A,Z3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y3),Z3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Z3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z4)) ) ) ) ).

% distrib_imp2
tff(fact_1053_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z4)) ) ).

% inf_sup_distrib1
tff(fact_1054_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z4: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z4),X)) ) ).

% inf_sup_distrib2
tff(fact_1055_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z4)) ) ).

% sup_inf_distrib1
tff(fact_1056_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Y: A,Z4: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z4),X)) ) ).

% sup_inf_distrib2
tff(fact_1057_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z4)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_1058_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z4)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_1059_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z4: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z4),X)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_1060_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Z4: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z4),X)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_1061_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) ) ).

% add_divide_distrib
tff(fact_1062_disjointI,axiom,
    ! [A: $tType,A3: set(A),B2: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A3))
         => ~ pp(aa(set(A),bool,member(A,X3),B2)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ).

% disjointI
tff(fact_1063_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A6))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),B5))
             => ( X4 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_1064_Int__empty__right,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_1065_Int__empty__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B5) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_1066_disjoint__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A6))
         => ~ pp(aa(set(A),bool,member(A,X4),B5)) ) ) ).

% disjoint_iff
tff(fact_1067_Int__emptyI,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => ~ pp(aa(set(A),bool,member(A,X3),B5)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_1068_Int__mono,axiom,
    ! [A: $tType,A6: set(A),C4: set(A),B5: set(A),D5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),D5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),D5))) ) ) ).

% Int_mono
tff(fact_1069_Int__lower1,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)),A6)) ).

% Int_lower1
tff(fact_1070_Int__lower2,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)),B5)) ).

% Int_lower2
tff(fact_1071_Int__absorb1,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = B5 ) ) ).

% Int_absorb1
tff(fact_1072_Int__absorb2,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = A6 ) ) ).

% Int_absorb2
tff(fact_1073_Int__greatest,axiom,
    ! [A: $tType,C4: set(A),A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5))) ) ) ).

% Int_greatest
tff(fact_1074_Int__Collect__mono,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),P2: fun(A,bool),Q2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => ( pp(aa(A,bool,P2,X3))
             => pp(aa(A,bool,Q2,X3)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(fun(A,bool),set(A),collect(A),P2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(fun(A,bool),set(A),collect(A),Q2)))) ) ) ).

% Int_Collect_mono
tff(fact_1075_inter__eq__subsetI,axiom,
    ! [A: $tType,S: set(A),S4: set(A),A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),S4))
     => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),S4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),S4) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_1076_Un__Int__crazy,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: 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)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A6)) = 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)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A6)) ).

% Un_Int_crazy
tff(fact_1077_Int__Un__distrib,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) = 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)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C4)) ).

% Int_Un_distrib
tff(fact_1078_Un__Int__distrib,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = 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)),sup_sup(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),C4)) ).

% Un_Int_distrib
tff(fact_1079_Int__Un__distrib2,axiom,
    ! [A: $tType,B5: set(A),C4: set(A),A6: 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)),sup_sup(set(A)),B5),C4)),A6) = 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)),B5),A6)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A6)) ).

% Int_Un_distrib2
tff(fact_1080_Un__Int__distrib2,axiom,
    ! [A: $tType,B5: set(A),C4: set(A),A6: 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)),B5),C4)),A6) = 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)),sup_sup(set(A)),B5),A6)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A6)) ).

% Un_Int_distrib2
tff(fact_1081_inf__option__def,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),Y) = case_option(option(A),A,none(A),aTP_Lamp_cs(option(A),fun(A,option(A)),Y),X) ) ).

% inf_option_def
tff(fact_1082_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z4)))) ) ).

% distrib_sup_le
tff(fact_1083_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z4: A] : pp(aa(A,bool,aa(A,fun(A,bool),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),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4)))) ) ).

% distrib_inf_le
tff(fact_1084_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ).

% power_mono
tff(fact_1085_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),N))) ) ) ).

% zero_le_power
tff(fact_1086_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),N))) ) ) ).

% zero_less_power
tff(fact_1087_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,power_power(A,A3),N))) ) ) ).

% one_le_power
tff(fact_1088_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1089_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))) ) ) ) ).

% divide_right_mono
tff(fact_1090_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1091_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1092_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1093_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1094_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1095_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3))) ) ) ) ).

% divide_right_mono_neg
tff(fact_1096_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_neg_neg
tff(fact_1097_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_1098_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_1099_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_pos_pos
tff(fact_1100_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_less_0_iff
tff(fact_1101_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( C3 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_1102_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_1103_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))) ) ) ) ).

% divide_strict_right_mono
tff(fact_1104_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1105_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: nat,N: nat] : aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),N)) ) ).

% power_add
tff(fact_1106_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,I2),M)),aa(nat,nat,power_power(nat,I2),N)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_1107_disjoint__mono,axiom,
    ! [A: $tType,A3: set(A),A4: set(A),B2: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B2),B3))
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B3) = bot_bot(set(A)) )
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B2) = bot_bot(set(A)) ) ) ) ) ).

% disjoint_mono
tff(fact_1108_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: 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)),A6),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A6)) ) ).

% Un_Int_assoc_eq
tff(fact_1109_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,B2),N)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_1110_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_1111_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_1112_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonpos_neg
tff(fact_1113_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% divide_nonneg_pos
tff(fact_1114_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_1115_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% divide_le_cancel
tff(fact_1116_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),Z4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z4)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).

% frac_less2
tff(fact_1117_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z4)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).

% frac_less
tff(fact_1118_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W2: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z4)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).

% frac_le
tff(fact_1119_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) ) ) ) ).

% div_positive
tff(fact_1120_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1121_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),N)))) ) ) ).

% power_gt1_lemma
tff(fact_1122_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),N)))) ) ) ).

% power_less_power_Suc
tff(fact_1123_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C3) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1124_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_1125_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_1126_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_1127_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_less_divide_eq
tff(fact_1128_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_divide_less_eq
tff(fact_1129_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_1130_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z4)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1131_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1132_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2))) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1133_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2))) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1134_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1135_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1136_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z4: A,A3: A,B2: A] :
          ( ( ( Z4 = 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),A3),Z4)),B2) = B2 ) )
          & ( ( Z4 != 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),A3),Z4)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z4))),Z4) ) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1137_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z4: A,A3: A,B2: A] :
          ( ( ( Z4 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z4)) = A3 ) )
          & ( ( Z4 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z4)),B2)),Z4) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1138_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z4: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z4 != 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),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z4)) = 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),X),Z4)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z4)) ) ) ) ) ).

% add_frac_eq
tff(fact_1139_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z4: 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),X),Y)),Z4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1140_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z4: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1141_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z4: A,X: A,Y: A] :
          ( ( Z4 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z4)) = 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),X),Z4)),Y)),Z4) ) ) ) ).

% add_divide_eq_iff
tff(fact_1142_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z4: A,X: A,Y: A] :
          ( ( Z4 != 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),X),Z4)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z4))),Z4) ) ) ) ).

% divide_add_eq_iff
tff(fact_1143_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1144_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,A3),N7))) ) ) ) ).

% power_strict_increasing
tff(fact_1145_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,A3),N7))) ) ) ) ).

% power_increasing
tff(fact_1146_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,power_power(A,zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1147_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2)) ) ) ).

% gt_half_sum
tff(fact_1148_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) ) ) ).

% less_half_sum
tff(fact_1149_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1150_Inf__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => lattic4895041142388067077er_set(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_1151_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),N))),aa(nat,A,power_power(A,A3),N))) ) ) ) ).

% power_Suc_less
tff(fact_1152_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2))) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1153_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z4: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1154_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z4)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1155_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_1156_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_divide_le_eq
tff(fact_1157_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_le_divide_eq
tff(fact_1158_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_1159_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2))) ) ) ) ) ).

% divide_left_mono
tff(fact_1160_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_1161_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_1162_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1163_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1164_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),N7)),aa(nat,A,power_power(A,A3),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1165_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N7: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N7)),aa(nat,A,power_power(A,A3),N))) ) ) ) ) ).

% power_decreasing
tff(fact_1166_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( ( aa(nat,A,power_power(A,A3),N) = aa(nat,A,power_power(A,B2),N) )
              <=> ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1167_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A3),N) = aa(nat,A,power_power(A,B2),N) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
               => ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1168_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1169_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,power_power(A,A3),N))) ) ) ) ).

% self_le_power
tff(fact_1170_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A3),N))) ) ) ) ).

% one_less_power
tff(fact_1171_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C3: 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),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self1
tff(fact_1172_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C3: 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),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self2
tff(fact_1173_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A3: 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),C3),B2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self3
tff(fact_1174_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A3: 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),C3)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% div_mult_self4
tff(fact_1175_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_1176_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1177_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1178_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))))) ) ).

% dividend_less_times_div
tff(fact_1179_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1180_merge__pure__and,axiom,
    ! [A3: bool,B2: bool] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn(A3)),pure_assn(B2)) = pure_assn(fconj(A3,B2)) ).

% merge_pure_and
tff(fact_1181_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_1182_inf__None__2,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [X: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),inf_inf(option(A)),X),none(A)) = none(A) ) ).

% inf_None_2
tff(fact_1183_mod__h__bot__iff_I6_J,axiom,
    ! [P2: assn,Q2: assn,H: heap_ext(product_unit)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_iff(6)
tff(fact_1184_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),S)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),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))),R2),S))) ) ).

% inf_Int_eq2
tff(fact_1185_mod__and__dist,axiom,
    ! [P2: assn,Q2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P2),Q2)),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),H)) ) ) ).

% mod_and_dist
tff(fact_1186_ent__conjI,axiom,
    ! [A6: assn,B5: assn,C4: assn] :
      ( entails(A6,B5)
     => ( entails(A6,C4)
       => entails(A6,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),B5),C4)) ) ) ).

% ent_conjI
tff(fact_1187_ent__conjE1,axiom,
    ! [A6: assn,C4: assn,B5: assn] :
      ( entails(A6,C4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A6),B5),C4) ) ).

% ent_conjE1
tff(fact_1188_ent__conjE2,axiom,
    ! [B5: assn,C4: assn,A6: assn] :
      ( entails(B5,C4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A6),B5),C4) ) ).

% ent_conjE2
tff(fact_1189_inf__Int__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),R2)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R2),S))) ) ).

% inf_Int_eq
tff(fact_1190_inf__set__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B5))) ).

% inf_set_def
tff(fact_1191_ex__distrib__and,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_ct(fun(A,assn),fun(assn,fun(A,assn)),P2),Q2)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,P2)),Q2) ).

% ex_distrib_and
tff(fact_1192_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M)) ).

% div_le_dividend
tff(fact_1193_div__le__mono,axiom,
    ! [M: nat,N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),K2))) ) ).

% div_le_mono
tff(fact_1194_preciseD,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn)),A3: A,P3: B,F4: assn,A4: A,F6: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( precise(A,B,R2)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A3),P3)),F4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A4),P3)),F6))),H))
       => ( A3 = A4 ) ) ) ).

% preciseD
tff(fact_1195_preciseI,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn))] :
      ( ! [A5: A,A8: A,H2: product_prod(heap_ext(product_unit),set(nat)),P5: B,F7: assn,F8: assn] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A5),P5)),F7)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A8),P5)),F8))),H2))
         => ( A5 = A8 ) )
     => precise(A,B,R2) ) ).

% preciseI
tff(fact_1196_precise__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,assn))] :
      ( precise(A,B,R2)
    <=> ! [A9: A,A10: A,H7: product_prod(heap_ext(product_unit),set(nat)),P6: B,F9: assn,F10: assn] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A9),P6)),F9)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R2,A10),P6)),F10))),H7))
         => ( A9 = A10 ) ) ) ).

% precise_def
tff(fact_1197_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1198_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),I2)) ) ).

% less_mult_imp_div_less
tff(fact_1199_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_1200_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))),M)) ).

% times_div_less_eq_dividend
tff(fact_1201_div__le__mono2,axiom,
    ! [M: nat,N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K2),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K2),M))) ) ) ).

% div_le_mono2
tff(fact_1202_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_1203_div__less__iff__less__mult,axiom,
    ! [Q5: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q5)),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5))) ) ) ).

% div_less_iff_less_mult
tff(fact_1204_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1205_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M)) ) ) ).

% div_less_dividend
tff(fact_1206_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: 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),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1207_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: 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),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1208_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q5: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q5)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1209_split__div,axiom,
    ! [P2: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P2,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I)),J) )
               => pp(aa(nat,bool,P2,I)) ) ) ) ) ) ).

% split_div
tff(fact_1210_wand__assnI,axiom,
    ! [H: heap_ext(product_unit),As: set(nat),Q2: assn,R2: assn] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
     => ( ! [H4: heap_ext(product_unit),As6: set(nat)] :
            ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As6) = bot_bot(set(nat)) )
           => ( relH(As,H,H4)
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As)))
               => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As6)))
                 => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(R2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6)))) ) ) ) )
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(wand_assn(Q2,R2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))) ) ) ).

% wand_assnI
tff(fact_1211_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb2))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ? [As12: set(nat),As23: set(nat)] :
                ( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
                & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
                & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As12)))
                & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As23))) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_1212_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb2))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ~ ? [As13: set(nat),As24: set(nat)] :
                  ( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
                  & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
                  & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As13)))
                  & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As24))) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_1213_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb2))
      <=> pp(Y) )
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ( pp(Y)
            <=> ~ ? [As1: set(nat),As22: set(nat)] :
                    ( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As1)))
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As22))) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_1214_times__assn__raw_Osimps,axiom,
    ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(P2,Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
    <=> ? [As1: set(nat),As22: set(nat)] :
          ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1)))
          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Q2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22))) ) ) ).

% times_assn_raw.simps
tff(fact_1215_update__wp__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),Y: A,X: A,Q2: fun(product_unit,assn)] : hoare_hoare_triple(product_unit,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,R3,Y)),wand_assn(sngr_assn(A,R3,X),aa(product_unit,assn,Q2,product_Unity))),ref_update(A,R3,X),Q2) ) ).

% update_wp_rule
tff(fact_1216_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb2))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
               => ? [As12: set(nat),As23: set(nat)] :
                    ( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As12)))
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As23))) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_1217_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb2))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
               => ~ ? [As13: set(nat),As24: set(nat)] :
                      ( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As13)))
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As24))) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_1218_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,times_assn_raw(X,Xa2),Xb2))
      <=> pp(Y) )
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( ( pp(Y)
                <=> ? [As1: set(nat),As22: set(nat)] :
                      ( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As1)))
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As22))) ) )
               => ~ pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_1219_unit__abs__eta__conv,axiom,
    ! [A: $tType,F3: fun(product_unit,A)] : aTP_Lamp_cu(fun(product_unit,A),fun(product_unit,A),F3) = F3 ).

% unit_abs_eta_conv
tff(fact_1220_in__range__empty,axiom,
    ! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ).

% in_range_empty
tff(fact_1221_in__range__dist__union,axiom,
    ! [H: heap_ext(product_unit),As: set(nat),As5: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As5))))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As5))) ) ) ).

% in_range_dist_union
tff(fact_1222_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),inf_inf(product_unit),Uu),Uv) = product_Unity ).

% inf_unit_def
tff(fact_1223_less__eq__assn__def,axiom,
    ! [A3: assn,B2: assn] :
      ( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less_eq(assn),A3),B2))
    <=> ( A3 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),B2) ) ) ).

% less_eq_assn_def
tff(fact_1224_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : Y = product_Unity ).

% old.unit.exhaust
tff(fact_1225_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),sup_sup(product_unit),Uu),Uv) = product_Unity ).

% sup_unit_def
tff(fact_1226_bot__unit__def,axiom,
    bot_bot(product_unit) = product_Unity ).

% bot_unit_def
tff(fact_1227_models__in__range,axiom,
    ! [P2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H))
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H)) ) ).

% models_in_range
tff(fact_1228_relH__in__rangeI_I2_J,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
      ( relH(As,H,H5)
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))) ) ).

% relH_in_rangeI(2)
tff(fact_1229_relH__in__rangeI_I1_J,axiom,
    ! [As: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit)] :
      ( relH(As,H,H5)
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))) ) ).

% relH_in_rangeI(1)
tff(fact_1230_relH__refl,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
     => relH(As,H,H) ) ).

% relH_refl
tff(fact_1231_in__range__subset,axiom,
    ! [As: set(nat),As5: set(nat),H: heap_ext(product_unit)] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As),As5))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As5)))
       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))) ) ) ).

% in_range_subset
tff(fact_1232_wait__def,axiom,
    ! [N: nat] : heap_Time_wait(N) = heap_Time_Heap2(product_unit,aTP_Lamp_cv(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N)) ).

% wait_def
tff(fact_1233_in__range_Osimps,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),As))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H))) ) ) ).

% in_range.simps
tff(fact_1234_in__range_Oelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
      <=> pp(Y) )
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ( pp(Y)
            <=> ~ ! [X4: nat] :
                    ( pp(aa(set(nat),bool,member(nat,X4),As2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H2))) ) ) ) ) ).

% in_range.elims(1)
tff(fact_1235_in__range_Oelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ~ ! [X5: nat] :
                  ( pp(aa(set(nat),bool,member(nat,X5),As2))
                 => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),lim(product_unit,H2))) ) ) ) ).

% in_range.elims(2)
tff(fact_1236_in__range_Oelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ! [X3: nat] :
                ( pp(aa(set(nat),bool,member(nat,X3),As2))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H2))) ) ) ) ).

% in_range.elims(3)
tff(fact_1237_old_Ounit_Orec,axiom,
    ! [T: $tType,F1: T] : product_rec_unit(T,F1,product_Unity) = F1 ).

% old.unit.rec
tff(fact_1238_wand__raw_Osimps,axiom,
    ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(P2,Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
        & ! [H8: heap_ext(product_unit),As7: set(nat)] :
            ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As),As7) = bot_bot(set(nat)) )
              & relH(As,H,H8)
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As)))
              & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
           => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Q2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7)))) ) ) ) ).

% wand_raw.simps
tff(fact_1239_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb2))
      <=> pp(Y) )
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ( pp(Y)
            <=> ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
                  & ! [H8: heap_ext(product_unit),As7: set(nat)] :
                      ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As7) = bot_bot(set(nat)) )
                        & relH(As2,H2,H8)
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As2)))
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
                     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As7)))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_1240_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb2))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
                & ! [H9: heap_ext(product_unit),As8: set(nat)] :
                    ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As8) = bot_bot(set(nat)) )
                      & relH(As2,H2,H9)
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As2)))
                      & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As8))) )
                   => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As8)))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_1241_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb2))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
              & ! [H4: heap_ext(product_unit),As6: set(nat)] :
                  ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As6) = bot_bot(set(nat)) )
                    & relH(As2,H2,H4)
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2)))
                    & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As6))) )
                 => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As6)))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_1242_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb2))
      <=> pp(Y) )
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( ( pp(Y)
                <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
                    & ! [H8: heap_ext(product_unit),As7: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As7) = bot_bot(set(nat)) )
                          & relH(As2,H2,H8)
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As2)))
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
                       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As7)))) ) ) )
               => ~ pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_1243_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb2))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
               => ~ ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
                    & ! [H9: heap_ext(product_unit),As8: set(nat)] :
                        ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As8) = bot_bot(set(nat)) )
                          & relH(As2,H2,H9)
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As2)))
                          & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As8))) )
                       => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As8)))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_1244_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,wand_raw(X,Xa2),Xb2))
     => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
               => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
                  & ! [H4: heap_ext(product_unit),As6: set(nat)] :
                      ( ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As2),As6) = bot_bot(set(nat)) )
                        & relH(As2,H2,H4)
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2)))
                        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As6))) )
                     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As6)))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_1245_in__range_Opelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
      <=> pp(Y) )
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( ( pp(Y)
                <=> ! [X4: nat] :
                      ( pp(aa(set(nat),bool,member(nat,X4),As2))
                     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H2))) ) )
               => ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))) ) ) ) ) ).

% in_range.pelims(1)
tff(fact_1246_less__assn__def,axiom,
    ! [A3: assn,B2: assn] :
      ( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less(assn),A3),B2))
    <=> ( pp(aa(assn,bool,aa(assn,fun(assn,bool),ord_less_eq(assn),A3),B2))
        & ( A3 != B2 ) ) ) ).

% less_assn_def
tff(fact_1247_in__range_Opelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
               => ! [X3: nat] :
                    ( pp(aa(set(nat),bool,member(nat,X3),As2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H2))) ) ) ) ) ) ).

% in_range.pelims(3)
tff(fact_1248_in__range_Opelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
               => ~ ! [X5: nat] :
                      ( pp(aa(set(nat),bool,member(nat,X5),As2))
                     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),lim(product_unit,H2))) ) ) ) ) ) ).

% in_range.pelims(2)
tff(fact_1249_execute__update,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),V2: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(product_unit,ref_update(A,R3,V2)),H) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,R3,V2,H)),one_one(nat)))) ) ).

% execute_update
tff(fact_1250_default__unit__def,axiom,
    default_default(product_unit) = product_Unity ).

% default_unit_def
tff(fact_1251_Ref__Time_Oupdate__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),V2: A] : ref_update(A,R3,V2) = heap_Time_heap(product_unit,aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_cw(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),R3),V2)) ) ).

% Ref_Time.update_def
tff(fact_1252_CODE__ABORT__def,axiom,
    ! [A: $tType,F3: fun(product_unit,A)] : cODE_ABORT(A,F3) = aa(product_unit,A,F3,product_Unity) ).

% CODE_ABORT_def
tff(fact_1253_verit__le__mono__div,axiom,
    ! [A6: nat,B5: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A6),B5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),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),A6),N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B5,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),N))) ) ) ).

% verit_le_mono_div
tff(fact_1254_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),X)),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1255_wand__assn__def,axiom,
    ! [P2: assn,Q2: assn] : wand_assn(P2,Q2) = abs_assn(wand_raw(rep_assn(P2),rep_assn(Q2))) ).

% wand_assn_def
tff(fact_1256_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,A3: A,B2: A,C3: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_1257_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M = N ) ) ) ).

% of_nat_eq_iff
tff(fact_1258_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_add_self1
tff(fact_1259_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_add_self2
tff(fact_1260_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = M ) ) ).

% mod_less
tff(fact_1261_Rep__assn__inverse,axiom,
    ! [X: assn] : abs_assn(rep_assn(X)) = X ).

% Rep_assn_inverse
tff(fact_1262_lim__set,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),V2: A,H: heap_ext(product_unit)] : lim(product_unit,ref_set(A,R3,V2,H)) = lim(product_unit,H) ) ).

% lim_set
tff(fact_1263_of__nat__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% of_nat_0
tff(fact_1264_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_1265_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_1266_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self1
tff(fact_1267_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self2
tff(fact_1268_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,B2: A,A3: 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),C3),B2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self3
tff(fact_1269_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A3: 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),C3)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self4
tff(fact_1270_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_iff
tff(fact_1271_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% of_nat_le_iff
tff(fact_1272_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_1273_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_1274_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_1275_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_1276_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_1277_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1278_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% of_nat_0_less_iff
tff(fact_1279_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,B2),W2)),X)) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1280_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,power_power(nat,B2),W2))) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1281_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,B2),W2))) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1282_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),W2)),X)) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1283_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C3) ) ).

% mod_add_right_eq
tff(fact_1284_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,C3)),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C3) ) ).

% mod_add_left_eq
tff(fact_1285_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,A4: A,B2: A,B3: A] :
          ( ( modulo_modulo(A,A3,C3) = modulo_modulo(A,A4,C3) )
         => ( ( modulo_modulo(A,B2,C3) = modulo_modulo(A,B3,C3) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3),C3) ) ) ) ) ).

% mod_add_cong
tff(fact_1286_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,C3)),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C3) ) ).

% mod_add_eq
tff(fact_1287_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_1288_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),M)) ).

% mod_less_eq_dividend
tff(fact_1289_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_1290_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(2)
tff(fact_1291_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% zle_int
tff(fact_1292_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(3)
tff(fact_1293_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z4: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z4)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),Z4) ).

% zadd_int_left
tff(fact_1294_int__ops_I5_J,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_1295_int__plus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_1296_int__ops_I7_J,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_1297_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_1298_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,N: nat] : modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_1299_nat__less__as__int,axiom,
    ! [X5: nat,Xa: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_less_as_int
tff(fact_1300_nat__leq__as__int,axiom,
    ! [X5: nat,Xa: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_leq_as_int
tff(fact_1301_Abs__assn__eqI_I1_J,axiom,
    ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Pr: assn] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P2,H2))
        <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Pr),H2)) )
     => ( abs_assn(P2) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_1302_Abs__assn__eqI_I2_J,axiom,
    ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Pr: assn] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P2,H2))
        <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Pr),H2)) )
     => ( Pr = abs_assn(P2) ) ) ).

% Abs_assn_eqI(2)
tff(fact_1303_pure__assn__def,axiom,
    ! [B2: bool] : pure_assn(B2) = abs_assn(pure_assn_raw(heap_ext(product_unit),nat,B2)) ).

% pure_assn_def
tff(fact_1304_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),modulo_modulo(A,A3,B2)),A3)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1305_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1306_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% of_nat_0_le_iff
tff(fact_1307_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_1308_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C3: A,B2: A] :
          ( ( modulo_modulo(A,A3,C3) = modulo_modulo(A,B2,C3) )
         => ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2)) ) ) ).

% mod_eqE
tff(fact_1309_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,C3)),modulo_modulo(A,B2,C3))),C3)) ) ).

% div_add1_eq
tff(fact_1310_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_1311_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_1312_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I2: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I2)),aa(nat,A,semiring_1_of_nat(A),J2))) ) ) ).

% of_nat_mono
tff(fact_1313_gcd__nat__induct,axiom,
    ! [P2: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [M5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M5),zero_zero(nat)))
     => ( ! [M5: nat,N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,N4),modulo_modulo(nat,M5,N4)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M5),N4)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_1314_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_less_divisor
tff(fact_1315_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J2: int,K2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K2)),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K2)),J2))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1316_pos__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ~ ! [N4: nat] :
            ( ( K2 = aa(nat,int,semiring_1_of_nat(int),N4) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4)) ) ) ).

% pos_int_cases
tff(fact_1317_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ? [N4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
          & ( K2 = aa(nat,int,semiring_1_of_nat(int),N4) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1318_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1319_bot__assn__def,axiom,
    bot_bot(assn) = abs_assn(aTP_Lamp_cx(product_prod(heap_ext(product_unit),set(nat)),bool)) ).

% bot_assn_def
tff(fact_1320_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A3,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1321_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( modulo_modulo(A,A3,B2) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1322_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),modulo_modulo(A,A3,B2)) = A3 ) ).

% mult_div_mod_eq
tff(fact_1323_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = A3 ) ).

% mod_mult_div_eq
tff(fact_1324_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = A3 ) ).

% mod_div_mult_eq
tff(fact_1325_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: 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),A3),B2)),B2)),modulo_modulo(A,A3,B2)) = A3 ) ).

% div_mult_mod_eq
tff(fact_1326_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2)) ) ).

% mod_div_decomp
tff(fact_1327_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),modulo_modulo(A,A3,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ).

% cancel_div_mod_rules(1)
tff(fact_1328_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A,C3: 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),A3),B2))),modulo_modulo(A,A3,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ).

% cancel_div_mod_rules(2)
tff(fact_1329_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C3))),C3)) ) ).

% div_mult1_eq
tff(fact_1330_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_le_divisor
tff(fact_1331_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
       => ? [Q4: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1332_mod__eq__nat2E,axiom,
    ! [M: nat,Q5: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q5) = modulo_modulo(nat,N,Q5) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ~ ! [S5: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),S5)) ) ) ).

% mod_eq_nat2E
tff(fact_1333_mod__eq__nat1E,axiom,
    ! [M: nat,Q5: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q5) = modulo_modulo(nat,N,Q5) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ~ ! [S5: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),S5)) ) ) ).

% mod_eq_nat1E
tff(fact_1334_div__less__mono,axiom,
    ! [A6: nat,B5: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A6),B5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( modulo_modulo(nat,A6,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B5,N) = zero_zero(nat) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A6),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),N))) ) ) ) ) ).

% div_less_mono
tff(fact_1335_div__mod__decomp,axiom,
    ! [A6: nat,N: nat] : A6 = 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),A6),N)),N)),modulo_modulo(nat,A6,N)) ).

% div_mod_decomp
tff(fact_1336_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q5: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N),Q5))),modulo_modulo(nat,M,N)) ).

% mod_mult2_eq
tff(fact_1337_sup__assn__def,axiom,
    ! [P2: assn,Q2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2) = abs_assn(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_cy(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P2),Q2)) ).

% sup_assn_def
tff(fact_1338_inf__assn__def,axiom,
    ! [P2: assn,Q2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P2),Q2) = abs_assn(aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_cz(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),P2),Q2)) ).

% inf_assn_def
tff(fact_1339_split__mod,axiom,
    ! [P2: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P2,modulo_modulo(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P2,M)) )
        & ( ( N != zero_zero(nat) )
         => ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I)),J) )
               => pp(aa(nat,bool,P2,J)) ) ) ) ) ) ).

% split_mod
tff(fact_1340_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
         => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = 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),A3),B2),C3))),modulo_modulo(A,A3,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1341_times__assn__def,axiom,
    ! [P2: assn,Q2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2) = abs_assn(times_assn_raw(rep_assn(P2),rep_assn(Q2))) ).

% times_assn_def
tff(fact_1342_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,C3: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),C3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_1343_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( 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),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_1344_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( 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),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_1345_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N4)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_1346_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_da(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_1347_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N4))) ) ).

% reals_Archimedean2
tff(fact_1348_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N4))) ) ).

% real_arch_simple
tff(fact_1349_relH__set__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),As: set(nat),H: heap_ext(product_unit),X: A] :
          ( ~ pp(aa(set(nat),bool,member(nat,addr_of_ref(A,R3)),As))
         => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
           => relH(As,H,ref_set(A,R3,X,H)) ) ) ) ).

% relH_set_ref
tff(fact_1350_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),N)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1351_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_1352_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_1353_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_eq
tff(fact_1354_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_1355_lessI,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,N))) ).

% lessI
tff(fact_1356_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% Suc_le_mono
tff(fact_1357_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc_right
tff(fact_1358_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_1359_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N))) ).

% zero_less_Suc
tff(fact_1360_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_1361_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_1362_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc_right
tff(fact_1363_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K2))
       => ( modulo_modulo(int,K2,L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1364_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),L))
       => ( modulo_modulo(int,K2,L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1365_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),L))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1366_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K2))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1367_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_1368_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ).

% one_le_mult_iff
tff(fact_1369_Suc__mod__mult__self4,axiom,
    ! [N: nat,K2: 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),N),K2)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self4
tff(fact_1370_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N: 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),K2),N)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self3
tff(fact_1371_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self2
tff(fact_1372_Suc__mod__mult__self1,axiom,
    ! [M: nat,K2: nat,N: 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),K2),N))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self1
tff(fact_1373_zle__add1__eq__le,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z4),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z4)) ) ).

% zle_add1_eq_le
tff(fact_1374_neg__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,A3,B2)),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),modulo_modulo(int,A3,B2))) ) ) ).

% neg_mod_conj
tff(fact_1375_pos__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A3,B2)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,A3,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_1376_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( modulo_modulo(int,I2,K2) = I2 )
    <=> ( ( K2 = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K2)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),I2)) ) ) ) ).

% zmod_trivial_iff
tff(fact_1377_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,M,K2)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_1378_zero__le__imp__eq__int,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ? [N4: nat] : K2 = aa(nat,int,semiring_1_of_nat(int),N4) ) ).

% zero_le_imp_eq_int
tff(fact_1379_nonneg__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ~ ! [N4: nat] : K2 != aa(nat,int,semiring_1_of_nat(int),N4) ) ).

% nonneg_int_cases
tff(fact_1380_le__imp__0__less,axiom,
    ! [Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z4))) ) ).

% le_imp_0_less
tff(fact_1381_int__one__le__iff__zero__less,axiom,
    ! [Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z4))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z4)) ) ).

% int_one_le_iff_zero_less
tff(fact_1382_zless__imp__add1__zle,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z4)) ) ).

% zless_imp_add1_zle
tff(fact_1383_add1__zle__eq,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z4))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4)) ) ).

% add1_zle_eq
tff(fact_1384_zle__iff__zadd,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z4))
    <=> ? [N3: nat] : Z4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),N3)) ) ).

% zle_iff_zadd
tff(fact_1385_incr__mult__lemma,axiom,
    ! [D3: int,P2: fun(int,bool),K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D3))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
         => ! [X5: int] :
              ( pp(aa(int,bool,P2,X5))
             => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1386_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R6: int,Q5: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R6)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R6))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q5),Q6)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1387_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R6: int,Q5: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R6)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R6))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R6),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q6),Q5)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1388_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q5: int,R3: int,B3: int,Q6: int,R6: 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)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R6) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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),B3),Q6)),R6)),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R6))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q6),Q5)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1389_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q5: int,R3: int,B3: int,Q6: int,R6: 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)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R6) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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),B3),Q6)),R6)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R6),B3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q5),Q6)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1390_int__mod__pos__eq,axiom,
    ! [A3: int,B2: int,Q5: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( modulo_modulo(int,A3,B2) = R3 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1391_int__mod__neg__eq,axiom,
    ! [A3: int,B2: int,Q5: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( modulo_modulo(int,A3,B2) = R3 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1392_q__pos__lemma,axiom,
    ! [B3: int,Q6: int,R6: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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),B3),Q6)),R6)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R6),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q6)) ) ) ) ).

% q_pos_lemma
tff(fact_1393_split__zmod,axiom,
    ! [P2: fun(int,bool),N: int,K2: int] :
      ( pp(aa(int,bool,P2,modulo_modulo(int,N,K2)))
    <=> ( ( ( K2 = zero_zero(int) )
         => pp(aa(int,bool,P2,N)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),K2))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,J)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,J)) ) ) ) ) ).

% split_zmod
tff(fact_1394_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,K2,L)),zero_zero(int))) ) ).

% neg_mod_sign
tff(fact_1395_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K2,L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1396_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)),zero_zero(int)))
       => ( modulo_modulo(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1397_zdiv__mono1,axiom,
    ! [A3: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2))) ) ) ).

% zdiv_mono1
tff(fact_1398_zdiv__mono2,axiom,
    ! [A3: int,B3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))) ) ) ) ).

% zdiv_mono2
tff(fact_1399_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I2),K2) = zero_zero(int) )
    <=> ( ( K2 = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K2)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),I2)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1400_zdiv__mono1__neg,axiom,
    ! [A3: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_1401_zdiv__mono2__neg,axiom,
    ! [A3: int,B3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1402_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L)))
    <=> ( ( K2 = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ) ).

% div_int_pos_iff
tff(fact_1403_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L))) ) ) ).

% div_positive_int
tff(fact_1404_div__nonneg__neg__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).

% div_nonneg_neg_le0
tff(fact_1405_div__nonpos__pos__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).

% div_nonpos_pos_le0
tff(fact_1406_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I2),K2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),I2)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1407_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1408_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1409_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
      <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1410_verit__le__mono__div__int,axiom,
    ! [A6: int,B5: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A6),B5))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),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),A6),N)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B5,N)),zero_zero(int)),one_one(int),zero_zero(int)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_1411_zmod__zmult2__eq,axiom,
    ! [C3: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C3))
     => ( modulo_modulo(int,A3,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = 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),A3),B2),C3))),modulo_modulo(int,A3,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1412_split__pos__lemma,axiom,
    ! [K2: int,P2: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P2,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K2)),modulo_modulo(int,N,K2)))
      <=> ! [I: int,J: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),K2))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P2,I),J)) ) ) ) ).

% split_pos_lemma
tff(fact_1413_split__neg__lemma,axiom,
    ! [K2: int,P2: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P2,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K2)),modulo_modulo(int,N,K2)))
      <=> ! [I: int,J: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),J))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),zero_zero(int)))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P2,I),J)) ) ) ) ).

% split_neg_lemma
tff(fact_1414_int__div__pos__eq,axiom,
    ! [A3: int,B2: int,Q5: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q5 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1415_int__div__neg__eq,axiom,
    ! [A3: int,B2: int,Q5: int,R3: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q5)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q5 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1416_split__zdiv,axiom,
    ! [P2: fun(int,bool),N: int,K2: int] :
      ( pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K2)))
    <=> ( ( ( K2 = zero_zero(int) )
         => pp(aa(int,bool,P2,zero_zero(int))) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),K2))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,I)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,I)) ) ) ) ) ).

% split_zdiv
tff(fact_1417_div__mod__decomp__int,axiom,
    ! [A6: int,N: int] : A6 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A6),N)),N)),modulo_modulo(int,A6,N)) ).

% div_mod_decomp_int
tff(fact_1418_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K2)),X)) ) ) ).

% int_div_less_self
tff(fact_1419_zdiv__mono__strict,axiom,
    ! [A6: int,B5: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A6),B5))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => ( ( modulo_modulo(int,A6,N) = zero_zero(int) )
         => ( ( modulo_modulo(int,B5,N) = zero_zero(int) )
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A6),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1420_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1421_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1422_div__neg__pos__less0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).

% div_neg_pos_less0
tff(fact_1423_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,K2,L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1424_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),modulo_modulo(int,K2,L))) ) ).

% neg_mod_bound
tff(fact_1425_zmult__zless__mono2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),J2))) ) ) ).

% zmult_zless_mono2
tff(fact_1426_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1427_int__Suc,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).

% int_Suc
tff(fact_1428_int__ops_I4_J,axiom,
    ! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).

% int_ops(4)
tff(fact_1429_int__if,axiom,
    ! [P2: bool,A3: nat,B2: nat] :
      ( ( pp(P2)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P2,A3,B2)) = aa(nat,int,semiring_1_of_nat(int),A3) ) )
      & ( ~ pp(P2)
       => ( aa(nat,int,semiring_1_of_nat(int),if(nat,P2,A3,B2)) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ) ).

% int_if
tff(fact_1430_nat__int__comparison_I1_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A3) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_1431_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4))
    <=> ? [N3: nat] : Z4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_1432_int__gr__induct,axiom,
    ! [K2: int,I2: int,P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),I2))
     => ( pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int))))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),I3))
             => ( pp(aa(int,bool,P2,I3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P2,I2)) ) ) ) ).

% int_gr_induct
tff(fact_1433_zless__add1__eq,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z4),one_one(int))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4))
        | ( W2 = Z4 ) ) ) ).

% zless_add1_eq
tff(fact_1434_odd__less__0__iff,axiom,
    ! [Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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)),Z4)),Z4)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),zero_zero(int))) ) ).

% odd_less_0_iff
tff(fact_1435_less__int__code_I1_J,axiom,
    ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),zero_zero(int))) ).

% less_int_code(1)
tff(fact_1436_plus__int__code_I1_J,axiom,
    ! [K2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),zero_zero(int)) = K2 ).

% plus_int_code(1)
tff(fact_1437_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_1438_odd__nonzero,axiom,
    ! [Z4: 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)),Z4)),Z4) != zero_zero(int) ).

% odd_nonzero
tff(fact_1439_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I2: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I2) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I2)) ) ).

% of_nat_aux.simps(2)
tff(fact_1440_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_1441_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_1442_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z22)) ).

% int_distrib(2)
tff(fact_1443_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W2)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W2)) ).

% int_distrib(1)
tff(fact_1444_zdiv__zmult2__eq,axiom,
    ! [C3: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C3))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),C3) ) ) ).

% zdiv_zmult2_eq
tff(fact_1445_verit__la__generic,axiom,
    ! [A3: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),X))
      | ( A3 = X )
      | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),A3)) ) ).

% verit_la_generic
tff(fact_1446_int__ge__induct,axiom,
    ! [K2: int,I2: int,P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),I2))
     => ( pp(aa(int,bool,P2,K2))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),I3))
             => ( pp(aa(int,bool,P2,I3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P2,I2)) ) ) ) ).

% int_ge_induct
tff(fact_1447_less__eq__int__code_I1_J,axiom,
    pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).

% less_eq_int_code(1)
tff(fact_1448_imp__le__cong,axiom,
    ! [X: int,X8: int,P2: bool,P4: bool] :
      ( ( X = X8 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
         => ( pp(P2)
          <=> pp(P4) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
           => pp(P2) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
           => pp(P4) ) ) ) ) ).

% imp_le_cong
tff(fact_1449_conj__le__cong,axiom,
    ! [X: int,X8: int,P2: bool,P4: bool] :
      ( ( X = X8 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
         => ( pp(P2)
          <=> pp(P4) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
            & pp(P2) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X8))
            & pp(P4) ) ) ) ) ).

% conj_le_cong
tff(fact_1450_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_1451_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_1452_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1453_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_1454_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1455_nat__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N4: nat] :
            ( pp(aa(nat,bool,P2,N4))
           => pp(aa(nat,bool,P2,aa(nat,nat,suc,N4))) )
       => pp(aa(nat,bool,P2,N)) ) ) ).

% nat_induct
tff(fact_1456_diff__induct,axiom,
    ! [P2: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,X3),zero_zero(nat)))
     => ( ! [Y3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,zero_zero(nat)),aa(nat,nat,suc,Y3)))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,X3),Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M),N)) ) ) ) ).

% diff_induct
tff(fact_1457_zero__induct,axiom,
    ! [P2: fun(nat,bool),K2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [N4: nat] :
            ( pp(aa(nat,bool,P2,aa(nat,nat,suc,N4)))
           => pp(aa(nat,bool,P2,N4)) )
       => pp(aa(nat,bool,P2,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_1458_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_1459_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_1460_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_1461_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).

% not0_implies_Suc
tff(fact_1462_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1463_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( ! [I3: nat] :
            ( ( J2 = aa(nat,nat,suc,I3) )
           => pp(aa(nat,bool,P2,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
             => ( pp(aa(nat,bool,P2,aa(nat,nat,suc,I3)))
               => pp(aa(nat,bool,P2,I3)) ) )
         => pp(aa(nat,bool,P2,I2)) ) ) ) ).

% strict_inc_induct
tff(fact_1464_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( ! [I3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I3),aa(nat,nat,suc,I3)))
       => ( ! [I3: nat,J3: nat,K: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),K))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I3),J3))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,J3),K))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I3),K)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I2),J2)) ) ) ) ).

% less_Suc_induct
tff(fact_1465_less__trans__Suc,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K2)) ) ) ).

% less_trans_Suc
tff(fact_1466_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_SucD
tff(fact_1467_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_1468_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
    <=> ? [M7: nat] :
          ( ( M = aa(nat,nat,suc,M7) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M7)) ) ) ).

% Suc_less_eq2
tff(fact_1469_All__less__Suc,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,N))
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
           => pp(aa(nat,bool,P2,I)) ) ) ) ).

% All_less_Suc
tff(fact_1470_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M))) ) ).

% not_less_eq
tff(fact_1471_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_1472_Ex__less__Suc,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,N))
        | ? [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
            & pp(aa(nat,bool,P2,I)) ) ) ) ).

% Ex_less_Suc
tff(fact_1473_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_1474_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_1475_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( ( aa(nat,nat,suc,M) != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_1476_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K2))
     => ~ ! [J3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J3))
           => ( K2 != aa(nat,nat,suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_1477_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_lessD
tff(fact_1478_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K2))
     => ( ( K2 != aa(nat,nat,suc,I2) )
       => ~ ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J3))
             => ( K2 != aa(nat,nat,suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_1479_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),X3))
       => ( ! [X3: nat,Y3: nat,Z3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),Y3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,Y3),Z3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),Z3)) ) )
         => ( ! [N4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,N4),aa(nat,nat,suc,N4)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1480_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,P2,M))
       => ( ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N4))
             => ( pp(aa(nat,bool,P2,N4))
               => pp(aa(nat,bool,P2,aa(nat,nat,suc,N4))) ) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_1481_full__nat__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( ! [N4: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M4)),N4))
             => pp(aa(nat,bool,P2,M4)) )
         => pp(aa(nat,bool,P2,N4)) )
     => pp(aa(nat,bool,P2,N)) ) ).

% full_nat_induct
tff(fact_1482_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_1483_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_1484_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        | ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_1485_Suc__le__D,axiom,
    ! [N: nat,M8: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M8))
     => ? [M5: nat] : M8 = aa(nat,nat,suc,M5) ) ).

% Suc_le_D
tff(fact_1486_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_1487_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_1488_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_leD
tff(fact_1489_nat__arith_Osuc1,axiom,
    ! [A6: nat,K2: nat,A3: nat] :
      ( ( A6 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),A3) )
     => ( aa(nat,nat,suc,A6) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,suc,A3)) ) ) ).

% nat_arith.suc1
tff(fact_1490_add__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc
tff(fact_1491_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_1492_Suc__mult__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_1493_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,M: nat] :
          ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N4)),aa(nat,A,F3,aa(nat,nat,suc,N4))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,M)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1494_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N2: nat] :
          ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N4)),aa(nat,A,F3,aa(nat,nat,suc,N4))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,N2))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_1495_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N2: nat] :
          ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N4))),aa(nat,A,F3,N4)))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,F3,N))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_1496_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N2: nat] :
          ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N4)),aa(nat,A,F3,aa(nat,nat,suc,N4))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,F3,N2))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_1497_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_1498_Ex__less__Suc2,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,zero_zero(nat)))
        | ? [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
            & pp(aa(nat,bool,P2,aa(nat,nat,suc,I))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1499_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% gr0_conv_Suc
tff(fact_1500_All__less__Suc2,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,zero_zero(nat)))
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
           => pp(aa(nat,bool,P2,aa(nat,nat,suc,I))) ) ) ) ).

% All_less_Suc2
tff(fact_1501_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).

% gr0_implies_Suc
tff(fact_1502_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( ( M = zero_zero(nat) )
        | ? [J: nat] :
            ( ( M = aa(nat,nat,suc,J) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1503_nat__compl__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N4: nat] :
            ( ! [Nn: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N4))
               => pp(aa(nat,bool,P2,Nn)) )
           => pp(aa(nat,bool,P2,aa(nat,nat,suc,N4))) )
       => pp(aa(nat,bool,P2,N)) ) ) ).

% nat_compl_induct
tff(fact_1504_nat__compl__induct_H,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N4: nat] :
            ( ! [Nn: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N4))
               => pp(aa(nat,bool,P2,Nn)) )
           => pp(aa(nat,bool,P2,aa(nat,nat,suc,N4))) )
       => pp(aa(nat,bool,P2,N)) ) ) ).

% nat_compl_induct'
tff(fact_1505_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_1506_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_1507_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N)) ) ).

% Suc_leI
tff(fact_1508_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_eq
tff(fact_1509_dec__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,P2,I2))
       => ( ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N4))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),J2))
               => ( pp(aa(nat,bool,P2,N4))
                 => pp(aa(nat,bool,P2,aa(nat,nat,suc,N4))) ) ) )
         => pp(aa(nat,bool,P2,J2)) ) ) ) ).

% dec_induct
tff(fact_1510_inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,P2,J2))
       => ( ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N4))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),J2))
               => ( pp(aa(nat,bool,P2,aa(nat,nat,suc,N4)))
                 => pp(aa(nat,bool,P2,N4)) ) ) )
         => pp(aa(nat,bool,P2,I2)) ) ) ) ).

% inc_induct
tff(fact_1511_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_lessD
tff(fact_1512_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1513_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_Suc_eq_le
tff(fact_1514_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_1515_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_1516_nat__in__between__eq_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),aa(nat,nat,suc,A3))) )
    <=> ( B2 = A3 ) ) ).

% nat_in_between_eq(2)
tff(fact_1517_nat__in__between__eq_I1_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(nat,nat,suc,A3))) )
    <=> ( B2 = aa(nat,nat,suc,A3) ) ) ).

% nat_in_between_eq(1)
tff(fact_1518_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ~ ! [Q4: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q4)) ) ).

% less_natE
tff(fact_1519_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)))) ).

% less_add_Suc1
tff(fact_1520_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)))) ).

% less_add_Suc2
tff(fact_1521_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1522_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ? [K: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)) ) ).

% less_imp_Suc_add
tff(fact_1523_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_1524_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1525_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_1526_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc
tff(fact_1527_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1528_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1529_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_1530_mod__induct,axiom,
    ! [P2: fun(nat,bool),N: nat,P3: nat,M: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),P3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),P3))
         => ( ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),P3))
               => ( pp(aa(nat,bool,P2,N4))
                 => pp(aa(nat,bool,P2,modulo_modulo(nat,aa(nat,nat,suc,N4),P3))) ) )
           => pp(aa(nat,bool,P2,M)) ) ) ) ) ).

% mod_induct
tff(fact_1531_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_1532_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),N))) ).

% Suc_div_le_mono
tff(fact_1533_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,N))),aa(nat,A,power_power(A,B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_1534_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A3),aa(nat,nat,suc,N)) = aa(nat,A,power_power(A,B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1535_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_1536_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I2: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I2) = I2 ) ).

% of_nat_aux.simps(1)
tff(fact_1537_ex__least__nat__less,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
       => ? [K: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),K))
               => ~ pp(aa(nat,bool,P2,I4)) )
            & pp(aa(nat,bool,P2,aa(nat,nat,suc,K))) ) ) ) ).

% ex_least_nat_less
tff(fact_1538_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_1539_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_1540_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% one_less_mult
tff(fact_1541_nat__induct__non__zero,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,P2,one_one(nat)))
       => ( ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
             => ( pp(aa(nat,bool,P2,N4))
               => pp(aa(nat,bool,P2,aa(nat,nat,suc,N4))) ) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1542_power__gt__expt,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(nat,nat,power_power(nat,N),K2))) ) ).

% power_gt_expt
tff(fact_1543_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,power_power(nat,I2),N))) ) ).

% nat_one_le_power
tff(fact_1544_relH__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [As: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: ref(A)] :
          ( relH(As,H,H5)
         => ( pp(aa(set(nat),bool,member(nat,addr_of_ref(A,R3)),As))
           => ( ref_get(A,H,R3) = ref_get(A,H5,R3) ) ) ) ) ).

% relH_ref
tff(fact_1545_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
         => ~ ! [N4: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),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,N4)))),E3)) ) ) ).

% nat_approx_posE
tff(fact_1546_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,N))),A3)) ) ) ) ).

% power_Suc_le_self
tff(fact_1547_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_1548_div__nat__eqI,axiom,
    ! [N: nat,Q5: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q5))))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = Q5 ) ) ) ).

% div_nat_eqI
tff(fact_1549_split__div_H,axiom,
    ! [P2: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P2,zero_zero(nat))) )
        | ? [Q7: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q7)),M))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q7))))
            & pp(aa(nat,bool,P2,Q7)) ) ) ) ).

% split_div'
tff(fact_1550_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X2: A] : aa(option(A),nat,size_option(A,X),aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1551_Heap_Osize__gen,axiom,
    ! [A: $tType,Xa2: fun(A,nat),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_size_Heap(A,Xa2,heap_Time_Heap2(A,X)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size_gen
tff(fact_1552_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(option(A),nat,size_option(A,X),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_1553_Heap_Osize_I2_J,axiom,
    ! [A: $tType,X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),heap_Time_Heap2(A,X)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size(2)
tff(fact_1554_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q5: int,R3: int] :
      ( eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3))
    <=> ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q5)),R3) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),L)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q5 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1555_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_1556_Heap_Osize__neq,axiom,
    ! [A: $tType,X: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),X) != zero_zero(nat) ).

% Heap.size_neq
tff(fact_1557_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P2: fun(int,fun(int,bool))] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
     => ( ! [I3: int,J3: int] :
            ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J3)))
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),J3))
               => pp(aa(int,bool,aa(int,fun(int,bool),P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J3)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P2,I3),J3)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P2,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_1558_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_1559_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_1560_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)),zero_zero(int)))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1561_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K2))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),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,aa(int,fun(int,int),minus_minus(int),K2),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1562_int__power__div__base,axiom,
    ! [M: nat,K2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,power_power(int,K2),M)),K2) = aa(nat,int,power_power(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1563_one__less__nat__eq,axiom,
    ! [Z4: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z4)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z4)) ) ).

% one_less_nat_eq
tff(fact_1564_Ref__Time_Opresent__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H: heap_ext(product_unit),R3: ref(A)] :
          ( ref_present(A,H,R3)
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,R3)),lim(product_unit,H))) ) ) ).

% Ref_Time.present_def
tff(fact_1565_verit__minus__simplify_I4_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,uminus_uminus(B),B2)) = B2 ) ).

% verit_minus_simplify(4)
tff(fact_1566_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% compl_le_compl_iff
tff(fact_1567_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% neg_le_iff_le
tff(fact_1568_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).

% add_diff_cancel_right'
tff(fact_1569_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_right
tff(fact_1570_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),A3) = B2 ) ).

% add_diff_cancel_left'
tff(fact_1571_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_left
tff(fact_1572_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ).

% diff_add_cancel
tff(fact_1573_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).

% add_diff_cancel
tff(fact_1574_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% compl_less_compl_iff
tff(fact_1575_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% neg_less_iff_less
tff(fact_1576_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_1577_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_1578_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_1579_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_1580_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1581_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% diff_Suc_Suc
tff(fact_1582_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N)),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),K2) ).

% Suc_diff_diff
tff(fact_1583_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_1584_diff__diff__left,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2)) ).

% diff_diff_left
tff(fact_1585_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1586_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1587_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1588_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1589_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_less_eq_nonneg
tff(fact_1590_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_1591_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_le_0_iff_le
tff(fact_1592_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_1593_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_1594_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1595_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1596_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_1597_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_pos
tff(fact_1598_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_1599_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_0_iff_less
tff(fact_1600_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_1601_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_1602_verit__minus__simplify_I3_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),B2) = aa(B,B,uminus_uminus(B),B2) ) ).

% verit_minus_simplify(3)
tff(fact_1603_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) ) ).

% diff_minus_eq_add
tff(fact_1604_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).

% uminus_add_conv_diff
tff(fact_1605_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_1606_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_1607_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% zero_less_diff
tff(fact_1608_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1609_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% diff_is_0_eq
tff(fact_1610_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J2) ) ) ).

% Nat.diff_diff_right
tff(fact_1611_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I2)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1612_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_1613_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_1614_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zle
tff(fact_1615_add__neg__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% add_neg_numeral_special(7)
tff(fact_1616_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = zero_zero(A) ) ) ).

% add_neg_numeral_special(8)
tff(fact_1617_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_1618_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,suc,J2)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1619_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2))),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1620_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_1621_nat__le__0,axiom,
    ! [Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),zero_zero(int)))
     => ( aa(int,nat,nat2,Z4) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_1622_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa(int,nat,nat2,I2) = zero_zero(nat) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_1623_zless__nat__conj,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z4)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z4))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4)) ) ) ).

% zless_nat_conj
tff(fact_1624_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zless
tff(fact_1625_int__nat__eq,axiom,
    ! [Z4: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z4)) = Z4 ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z4)) = zero_zero(int) ) ) ) ).

% int_nat_eq
tff(fact_1626_zle__diff1__eq,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z4),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4)) ) ).

% zle_diff1_eq
tff(fact_1627_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_1628_zero__less__nat__eq,axiom,
    ! [Z4: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z4)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z4)) ) ).

% zero_less_nat_eq
tff(fact_1629_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_1630_diff__commute,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),K2)),J2) ).

% diff_commute
tff(fact_1631_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_1632_nat__diff__distrib,axiom,
    ! [Z7: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z7))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z7),Z4))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z4),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z4)),aa(int,nat,nat2,Z7)) ) ) ) ).

% nat_diff_distrib
tff(fact_1633_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B5: A,K2: A,B2: A,A3: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1634_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_1635_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1636_int__minus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% int_minus
tff(fact_1637_nat__minus__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X5),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_1638_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_1639_diff__nat__eq__if,axiom,
    ! [Z7: int,Z4: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z7),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z4)),aa(int,nat,nat2,Z7)) = aa(int,nat,nat2,Z4) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z7),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z4)),aa(int,nat,nat2,Z7)) = if(nat,aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z4),Z7)),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z4),Z7))) ) ) ) ).

% diff_nat_eq_if
tff(fact_1640_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1641_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3))) ) ) ).

% diff_right_mono
tff(fact_1642_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))) ) ) ).

% diff_left_mono
tff(fact_1643_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).

% diff_mono
tff(fact_1644_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3))) ) ) ).

% diff_strict_right_mono
tff(fact_1645_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))) ) ) ).

% diff_strict_left_mono
tff(fact_1646_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3)) ) ) ) ).

% diff_eq_diff_less
tff(fact_1647_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).

% diff_strict_mono
tff(fact_1648_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% diff_diff_eq
tff(fact_1649_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C3: A,B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) = A3 )
         => ( C3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ) ) ).

% add_implies_diff
tff(fact_1650_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1651_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2) ) ).

% diff_add_eq
tff(fact_1652_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2) ) ).

% diff_diff_eq2
tff(fact_1653_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) ) ).

% add_diff_eq
tff(fact_1654_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C3 ) ) ) ).

% eq_diff_eq
tff(fact_1655_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = C3 )
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1656_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A6: A,K2: A,A3: A,B2: A] :
          ( ( A6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A3) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A6),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1657_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P2: fun(A,bool),D5: A,Q2: fun(A,bool)] :
          ( ! [X3: A,K: A] :
              ( pp(aa(A,bool,P2,X3))
            <=> pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
         => ( ! [X3: A,K: A] :
                ( pp(aa(A,bool,Q2,X3))
              <=> pp(aa(A,bool,Q2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
           => ! [X5: A,K4: A] :
                ( ( pp(aa(A,bool,P2,X5))
                  | pp(aa(A,bool,Q2,X5)) )
              <=> ( pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))))
                  | pp(aa(A,bool,Q2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1658_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P2: fun(A,bool),D5: A,Q2: fun(A,bool)] :
          ( ! [X3: A,K: A] :
              ( pp(aa(A,bool,P2,X3))
            <=> pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
         => ( ! [X3: A,K: A] :
                ( pp(aa(A,bool,Q2,X3))
              <=> pp(aa(A,bool,Q2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
           => ! [X5: A,K4: A] :
                ( ( pp(aa(A,bool,P2,X5))
                  & pp(aa(A,bool,Q2,X5)) )
              <=> ( pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))))
                  & pp(aa(A,bool,Q2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1659_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_le_swap2
tff(fact_1660_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_le_swap1
tff(fact_1661_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_1662_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_imp_neg_le
tff(fact_1663_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_le_iff
tff(fact_1664_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_minus_iff
tff(fact_1665_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_less_swap2
tff(fact_1666_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_less_swap1
tff(fact_1667_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_less_iff
tff(fact_1668_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% less_minus_iff
tff(fact_1669_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% verit_negate_coefficient(2)
tff(fact_1670_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ).

% is_num_normalize(8)
tff(fact_1671_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ).

% add.inverse_distrib_swap
tff(fact_1672_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A6: A,K2: A,A3: A] :
          ( ( A6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A3) )
         => ( aa(A,A,uminus_uminus(A),A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_1673_length__induct,axiom,
    ! [A: $tType,P2: fun(list(A),bool),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs2)))
             => pp(aa(list(A),bool,P2,Ys)) )
         => pp(aa(list(A),bool,P2,Xs2)) )
     => pp(aa(list(A),bool,P2,Xs)) ) ).

% length_induct
tff(fact_1674_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1675_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) = zero_zero(nat) )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_1676_zero__induct__lemma,axiom,
    ! [P2: fun(nat,bool),K2: nat,I2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [N4: nat] :
            ( pp(aa(nat,bool,P2,aa(nat,nat,suc,N4)))
           => pp(aa(nat,bool,P2,N4)) )
       => pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),I2))) ) ) ).

% zero_induct_lemma
tff(fact_1677_less__imp__diff__less,axiom,
    ! [J2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),N)),K2)) ) ).

% less_imp_diff_less
tff(fact_1678_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ) ).

% diff_less_mono2
tff(fact_1679_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ).

% diff_le_mono2
tff(fact_1680_le__diff__iff_H,axiom,
    ! [A3: nat,C3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),C3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),C3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),A3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),B2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A3)) ) ) ) ).

% le_diff_iff'
tff(fact_1681_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ).

% diff_le_self
tff(fact_1682_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L))) ) ).

% diff_le_mono
tff(fact_1683_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1684_le__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% le_diff_iff
tff(fact_1685_eq__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1686_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% Nat.diff_cancel
tff(fact_1687_diff__cancel2,axiom,
    ! [M: nat,K2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% diff_cancel2
tff(fact_1688_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).

% diff_add_inverse
tff(fact_1689_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).

% diff_add_inverse2
tff(fact_1690_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)) ).

% diff_mult_distrib
tff(fact_1691_diff__mult__distrib2,axiom,
    ! [K2: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) ).

% diff_mult_distrib2
tff(fact_1692_int__ops_I6_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = zero_zero(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ) ) ).

% int_ops(6)
tff(fact_1693_nat__mult__distrib__neg,axiom,
    ! [Z4: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),zero_zero(int)))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z4),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z4))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z7))) ) ) ).

% nat_mult_distrib_neg
tff(fact_1694_zdiff__int__split,axiom,
    ! [P2: fun(int,bool),X: nat,Y: nat] :
      ( pp(aa(int,bool,P2,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Y))))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
         => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y)))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
         => pp(aa(int,bool,P2,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_1695_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).

% le_iff_diff_le_0
tff(fact_1696_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_1697_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).

% less_iff_diff_less_0
tff(fact_1698_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% diff_le_eq
tff(fact_1699_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3)) ) ) ).

% le_diff_eq
tff(fact_1700_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),A3) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1701_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A3))) ) ) ).

% le_add_diff
tff(fact_1702_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),B2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1703_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1704_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1705_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1706_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1707_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1708_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1709_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1710_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K2: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K2))) ) ) ).

% add_le_imp_le_diff
tff(fact_1711_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K2: A,N: A,J2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K2)),J2)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1712_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% diff_shunt_var
tff(fact_1713_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3)) ) ) ).

% less_diff_eq
tff(fact_1714_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% diff_less_eq
tff(fact_1715_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1716_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( 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),minus_minus(A),A3),B2)),E3)),C3) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_1717_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( C3 = 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),minus_minus(A),B2),A3)),E3)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_1718_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),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),X),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) ) ).

% square_diff_square_factored
tff(fact_1719_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y))) ) ).

% nat_mono
tff(fact_1720_ex__nat,axiom,
    ! [P2: fun(nat,bool)] :
      ( ? [X_12: nat] : pp(aa(nat,bool,P2,X_12))
    <=> ? [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
          & pp(aa(nat,bool,P2,aa(int,nat,nat2,X4))) ) ) ).

% ex_nat
tff(fact_1721_all__nat,axiom,
    ! [P2: fun(nat,bool)] :
      ( ! [X_12: nat] : pp(aa(nat,bool,P2,X_12))
    <=> ! [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
         => pp(aa(nat,bool,P2,aa(int,nat,nat2,X4))) ) ) ).

% all_nat
tff(fact_1722_eq__nat__nat__iff,axiom,
    ! [Z4: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z7))
       => ( ( aa(int,nat,nat2,Z4) = aa(int,nat,nat2,Z7) )
        <=> ( Z4 = Z7 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_1723_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K2),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_1724_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_1725_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_1726_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_1727_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_1728_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_1729_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_1730_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% le_minus_one_simps(2)
tff(fact_1731_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(4)
tff(fact_1732_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_1733_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% less_minus_one_simps(2)
tff(fact_1734_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(4)
tff(fact_1735_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,suc,N) = M )
     => ( N = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_1736_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ) ) ).

% diff_less
tff(fact_1737_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ).

% Suc_diff_Suc
tff(fact_1738_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(nat,nat,suc,M))) ).

% diff_less_Suc
tff(fact_1739_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) ) ).

% Suc_diff_le
tff(fact_1740_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1741_less__diff__iff,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% less_diff_iff
tff(fact_1742_diff__less__mono,axiom,
    ! [A3: nat,B2: nat,C3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C3),A3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),C3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C3))) ) ) ).

% diff_less_mono
tff(fact_1743_less__diff__conv,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J2)) ) ).

% less_diff_conv
tff(fact_1744_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1745_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2) = K2 )
      <=> ( J2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1746_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1747_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1748_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J2)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1749_le__diff__conv,axiom,
    ! [J2: nat,K2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2))) ) ).

% le_diff_conv
tff(fact_1750_int__le__induct,axiom,
    ! [I2: int,K2: int,P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),K2))
     => ( pp(aa(int,bool,P2,K2))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K2))
             => ( pp(aa(int,bool,P2,I3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P2,I2)) ) ) ) ).

% int_le_induct
tff(fact_1751_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1752_int__less__induct,axiom,
    ! [I2: int,K2: int,P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K2))
     => ( pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),one_one(int))))
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I3),K2))
             => ( pp(aa(int,bool,P2,I3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P2,I2)) ) ) ) ).

% int_less_induct
tff(fact_1753_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ).

% mod_geq
tff(fact_1754_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = M ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ) ).

% mod_if
tff(fact_1755_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N) ) ) ).

% le_mod_geq
tff(fact_1756_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% not_int_zless_negative
tff(fact_1757_option_Osize__neq,axiom,
    ! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_1758_le__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C3)),D3)) ) ) ).

% le_add_iff1
tff(fact_1759_le__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),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),minus_minus(A),B2),A3)),E3)),D3))) ) ) ).

% le_add_iff2
tff(fact_1760_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),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),minus_minus(A),B2),A3)),E3)),D3))) ) ) ).

% less_add_iff2
tff(fact_1761_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C3)),D3)) ) ) ).

% less_add_iff1
tff(fact_1762_nat__mono__iff,axiom,
    ! [Z4: int,W2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z4)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4)) ) ) ).

% nat_mono_iff
tff(fact_1763_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),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),X),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1764_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% le_minus_one_simps(1)
tff(fact_1765_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(3)
tff(fact_1766_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z4: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(int,nat,nat2,Z4)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z4)) ) ).

% zless_nat_eq_int_zless
tff(fact_1767_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% less_minus_one_simps(1)
tff(fact_1768_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(3)
tff(fact_1769_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_le_iff
tff(fact_1770_int__eq__iff,axiom,
    ! [M: nat,Z4: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z4 )
    <=> ( ( M = aa(int,nat,nat2,Z4) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4)) ) ) ).

% int_eq_iff
tff(fact_1771_nat__0__le,axiom,
    ! [Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z4)) = Z4 ) ) ).

% nat_0_le
tff(fact_1772_nat__int__add,axiom,
    ! [A3: nat,B2: nat] : aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2) ).

% nat_int_add
tff(fact_1773_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% inf_shunt
tff(fact_1774_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z4))) ) ) ).

% shunt1
tff(fact_1775_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z4))) ) ) ).

% shunt2
tff(fact_1776_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q5: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q5),R3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q5))),R3)) ) ) ).

% sup_neg_inf
tff(fact_1777_nat__power__eq,axiom,
    ! [Z4: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( aa(int,nat,nat2,aa(nat,int,power_power(int,Z4),N)) = aa(nat,nat,power_power(nat,aa(int,nat,nat2,Z4)),N) ) ) ).

% nat_power_eq
tff(fact_1778_diff__Suc__less,axiom,
    ! [N: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,I2))),N)) ) ).

% diff_Suc_less
tff(fact_1779_nat__diff__split__asm,axiom,
    ! [P2: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
            & ~ pp(aa(nat,bool,P2,zero_zero(nat))) )
          | ? [D6: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
              & ~ pp(aa(nat,bool,P2,D6)) ) ) ) ).

% nat_diff_split_asm
tff(fact_1780_nat__diff__split,axiom,
    ! [P2: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
         => pp(aa(nat,bool,P2,zero_zero(nat))) )
        & ! [D6: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
           => pp(aa(nat,bool,P2,D6)) ) ) ) ).

% nat_diff_split
tff(fact_1781_less__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2))) ) ) ).

% less_diff_conv2
tff(fact_1782_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,bool),P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K: int] :
            ( pp(aa(int,bool,P1,X3))
          <=> pp(aa(int,bool,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) )
       => ( ? [Z6: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z6))
             => ( pp(aa(int,bool,P2,X3))
              <=> pp(aa(int,bool,P1,X3)) ) )
         => ( ? [X_13: int] : pp(aa(int,bool,P1,X_13))
           => ? [X_1: int] : pp(aa(int,bool,P2,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_1783_plusinfinity,axiom,
    ! [D3: int,P4: fun(int,bool),P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K: int] :
            ( pp(aa(int,bool,P4,X3))
          <=> pp(aa(int,bool,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) )
       => ( ? [Z6: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z6),X3))
             => ( pp(aa(int,bool,P2,X3))
              <=> pp(aa(int,bool,P4,X3)) ) )
         => ( ? [X_13: int] : pp(aa(int,bool,P4,X_13))
           => ? [X_1: int] : pp(aa(int,bool,P2,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_1784_int__cases4,axiom,
    ! [M: int] :
      ( ! [N4: nat] : M != aa(nat,int,semiring_1_of_nat(int),N4)
     => ~ ! [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N4)) ) ) ) ).

% int_cases4
tff(fact_1785_nat__eq__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1786_nat__eq__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1787_nat__le__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1788_nat__le__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1789_nat__diff__add__eq1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( aa(nat,nat,aa(nat,fun(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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)) = aa(nat,nat,aa(nat,fun(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,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1790_nat__diff__add__eq2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( aa(nat,nat,aa(nat,fun(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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)) = aa(nat,nat,aa(nat,fun(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,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1791_int__induct,axiom,
    ! [P2: fun(int,bool),K2: int,I2: int] :
      ( pp(aa(int,bool,P2,K2))
     => ( ! [I3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),I3))
           => ( pp(aa(int,bool,P2,I3))
             => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int)))) ) )
       => ( ! [I3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I3),K2))
             => ( pp(aa(int,bool,P2,I3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int)))) ) )
         => pp(aa(int,bool,P2,I2)) ) ) ) ).

% int_induct
tff(fact_1792_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_1793_nonpos__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),zero_zero(int)))
     => ~ ! [N4: nat] : K2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N4)) ) ).

% nonpos_int_cases
tff(fact_1794_negative__zle__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int))) ).

% negative_zle_0
tff(fact_1795_nat__plus__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X5),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_1796_nat__times__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X5),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_1797_nat__div__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X5),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_1798_nat__mod__as__int,axiom,
    ! [X5: nat,Xa: nat] : modulo_modulo(nat,X5,Xa) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X5),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_1799_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z4: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z4 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z4)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z4)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z4))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_1800_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z4: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z4 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z4)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z4)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z4))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_1801_nat__less__eq__zless,axiom,
    ! [W2: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z4)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4)) ) ) ).

% nat_less_eq_zless
tff(fact_1802_nat__le__eq__zle,axiom,
    ! [W2: int,Z4: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W2))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z4)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z4)) ) ) ).

% nat_le_eq_zle
tff(fact_1803_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa(int,nat,nat2,W2) = M )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_1804_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa(int,nat,nat2,W2) )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_1805_split__nat,axiom,
    ! [P2: fun(nat,bool),I2: int] :
      ( pp(aa(nat,bool,P2,aa(int,nat,nat2,I2)))
    <=> ( ! [N3: nat] :
            ( ( I2 = aa(nat,int,semiring_1_of_nat(int),N3) )
           => pp(aa(nat,bool,P2,N3)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
         => pp(aa(nat,bool,P2,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_1806_le__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(int,nat,nat2,K2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),K2)) ) ) ).

% le_nat_iff
tff(fact_1807_nat__add__distrib,axiom,
    ! [Z4: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z7))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z4),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z4)),aa(int,nat,nat2,Z7)) ) ) ) ).

% nat_add_distrib
tff(fact_1808_nat__mult__distrib,axiom,
    ! [Z4: int,Z7: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z4),Z7)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z4)),aa(int,nat,nat2,Z7)) ) ) ).

% nat_mult_distrib
tff(fact_1809_Suc__as__int,axiom,
    ! [X5: nat] : aa(nat,nat,suc,X5) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X5)),one_one(int))) ).

% Suc_as_int
tff(fact_1810_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1811_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1812_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1813_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1814_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_1815_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_1816_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z4: A,X: A,Y: A] :
          ( ( Z4 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z4))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z4))),Z4) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1817_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z4: A,A3: A,B2: A] :
          ( ( ( Z4 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z4))),B2) = B2 ) )
          & ( ( Z4 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z4))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z4))),Z4) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1818_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_1819_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_1820_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => ( aa(int,nat,nat2,modulo_modulo(int,X,Y)) = modulo_modulo(nat,aa(int,nat,nat2,X),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_1821_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),N)) ) ) ) ) ).

% power_diff
tff(fact_1822_nz__le__conv__less,axiom,
    ! [K2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),aa(nat,nat,suc,zero_zero(nat)))),M)) ) ) ).

% nz_le_conv_less
tff(fact_1823_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1824_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1825_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_1826_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),M))
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1827_nat__less__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_1828_nat__less__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_1829_decr__mult__lemma,axiom,
    ! [D3: int,P2: fun(int,bool),K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D3))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
         => ! [X5: int] :
              ( pp(aa(int,bool,P2,X5))
             => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1830_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).

% div_if
tff(fact_1831_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).

% div_geq
tff(fact_1832_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero(int) )
     => ( ! [N4: nat] :
            ( ( K2 = aa(nat,int,semiring_1_of_nat(int),N4) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4)) )
       => ~ ! [N4: nat] :
              ( ( K2 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N4)) )
             => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4)) ) ) ) ).

% int_cases3
tff(fact_1833_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_1834_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))) ).

% not_zle_0_negative
tff(fact_1835_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N4: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N4))) ) ).

% negD
tff(fact_1836_negative__zless__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_1837_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K2))
       => ( modulo_modulo(int,K2,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1838_verit__less__mono__div__int2,axiom,
    ! [A6: int,B5: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A6),B5))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A6),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_1839_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1840_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R3: A,S2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R3),S2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(A,A,aa(A,fun(A,A),minus_minus(A),V2),U))),S2))),V2)) ) ) ) ) ).

% scaling_mono
tff(fact_1841_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z4)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z4)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_1842_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),M))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).

% nat_less_iff
tff(fact_1843_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_1844_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_1845_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_1846_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_1847_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_1848_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_1849_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),N)) = aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_1850_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A3) = aa(nat,A,power_power(A,A3),N) ) ) ) ).

% power_minus_mult
tff(fact_1851_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_1852_neg__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
     => ~ ! [N4: nat] :
            ( ( K2 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N4)) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4)) ) ) ).

% neg_int_cases
tff(fact_1853_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1854_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_db(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_1855_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_1856_normalize__negative,axiom,
    ! [Q5: int,P3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q5),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q5)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q5))) ) ) ).

% normalize_negative
tff(fact_1857_length__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Xs: list(A)] : hoare_hoare_triple(nat,snga_assn(A,A3,Xs),array_len(A,A3),aa(list(A),fun(nat,assn),aTP_Lamp_dc(array(A),fun(list(A),fun(nat,assn)),A3),Xs)) ) ).

% length_rule
tff(fact_1858_Diff__cancel,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),A6) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1859_empty__Diff,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A6) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1860_Diff__empty,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),bot_bot(set(A))) = A6 ).

% Diff_empty
tff(fact_1861_Compl__anti__mono,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B5)),aa(set(A),set(A),uminus_uminus(set(A)),A6))) ) ).

% Compl_anti_mono
tff(fact_1862_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)),aa(set(A),set(A),uminus_uminus(set(A)),B5)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6)) ) ).

% Compl_subset_Compl_iff
tff(fact_1863_Diff__Compl,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) ).

% Diff_Compl
tff(fact_1864_inter__compl__diff__conv,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) ).

% inter_compl_diff_conv
tff(fact_1865_Compl__Diff__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)),B5) ).

% Compl_Diff_eq
tff(fact_1866_Un__Diff__cancel,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) ).

% Un_Diff_cancel
tff(fact_1867_Un__Diff__cancel2,axiom,
    ! [A: $tType,B5: set(A),A6: 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)),minus_minus(set(A)),B5),A6)),A6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A6) ).

% Un_Diff_cancel2
tff(fact_1868_diff__diff__add__mset,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),P2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),N7)),P2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(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),P2)) ).

% diff_diff_add_mset
tff(fact_1869_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_1870_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_1871_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType,X: 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)),X),Y) )
    <=> ( ( X = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_1872_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Y) = zero_zero(multiset(A)) )
    <=> ( ( X = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_1873_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ).

% Diff_eq_empty_iff
tff(fact_1874_Diff__disjoint,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_1875_Compl__disjoint,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),uminus_uminus(set(A)),A6)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_1876_Compl__disjoint2,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)),A6) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_1877_mod__not__dist,axiom,
    ! [P2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,uminus_uminus(assn),P2)),H))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H))
        & ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H)) ) ) ).

% mod_not_dist
tff(fact_1878_minus__assn__def,axiom,
    ! [A3: assn,B2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),minus_minus(assn),A3),B2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A3),aa(assn,assn,uminus_uminus(assn),B2)) ).

% minus_assn_def
tff(fact_1879_Multiset_Odiff__add,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),Q2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(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),Q2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),N7)),Q2) ).

% Multiset.diff_add
tff(fact_1880_diff__union__cancelL,axiom,
    ! [A: $tType,N7: multiset(A),M6: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(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_1881_diff__union__cancelR,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(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_1882_union__diff__assoc,axiom,
    ! [A: $tType,C4: multiset(A),B5: multiset(A),A6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),B5) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),A6),B5)),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C4)) ) ) ).

% union_diff_assoc
tff(fact_1883_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: multiset(A),D5: multiset(A),C4: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B5),D5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D5),C4))) ) ) ).

% union_le_mono1
tff(fact_1884_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: multiset(A),D5: multiset(A),C4: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B5),D5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D5))) ) ) ).

% union_le_mono2
tff(fact_1885_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: multiset(A),C4: multiset(A),B5: multiset(A),D5: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),A6),C4))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B5),D5))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D5))) ) ) ) ).

% union_less_mono
tff(fact_1886_empty__neutral_I1_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),zero_zero(multiset(A))),X) = X ).

% empty_neutral(1)
tff(fact_1887_empty__neutral_I2_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),zero_zero(multiset(A))) = X ).

% empty_neutral(2)
tff(fact_1888_mset__distrib,axiom,
    ! [A: $tType,A6: multiset(A),B5: 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)),A6),B5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7) )
     => ~ ! [Am: multiset(A),An: multiset(A)] :
            ( ( A6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
           => ! [Bm: multiset(A),Bn: multiset(A)] :
                ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
               => ( ( M6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
                 => ( N7 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_1889_multi__union__self__other__eq,axiom,
    ! [A: $tType,A6: multiset(A),X6: multiset(A),Y6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),X6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),Y6) )
     => ( X6 = Y6 ) ) ).

% multi_union_self_other_eq
tff(fact_1890_union__right__cancel,axiom,
    ! [A: $tType,M6: multiset(A),K5: multiset(A),N7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),K5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N7),K5) )
    <=> ( M6 = N7 ) ) ).

% union_right_cancel
tff(fact_1891_union__left__cancel,axiom,
    ! [A: $tType,K5: 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)),K5),M6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K5),N7) )
    <=> ( M6 = N7 ) ) ).

% union_left_cancel
tff(fact_1892_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_1893_union__lcomm,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),K5: 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),K5)) = 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),K5)) ).

% union_lcomm
tff(fact_1894_union__assoc,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),K5: 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)),K5) = 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),K5)) ).

% union_assoc
tff(fact_1895_Diff__mono,axiom,
    ! [A: $tType,A6: set(A),C4: set(A),D5: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D5),B5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),D5))) ) ) ).

% Diff_mono
tff(fact_1896_Diff__subset,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),A6)) ).

% Diff_subset
tff(fact_1897_double__diff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),A6)) = A6 ) ) ) ).

% double_diff
tff(fact_1898_Diff__Int__distrib2,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: 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)),minus_minus(set(A)),A6),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ).

% Diff_Int_distrib2
tff(fact_1899_Diff__Int__distrib,axiom,
    ! [A: $tType,C4: set(A),A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A6)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),B5)) ).

% Diff_Int_distrib
tff(fact_1900_Diff__Diff__Int,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) ).

% Diff_Diff_Int
tff(fact_1901_Diff__Int2,axiom,
    ! [A: $tType,A6: set(A),C4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C4)),B5) ).

% Diff_Int2
tff(fact_1902_Int__Diff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C4)) ).

% Int_Diff
tff(fact_1903_Un__Diff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)),C4) = 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)),minus_minus(set(A)),A6),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C4)) ).

% Un_Diff
tff(fact_1904_set__diff__diff__left,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) ).

% set_diff_diff_left
tff(fact_1905_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
     => ? [B4: A] : pp(aa(set(A),bool,member(A,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6))) ) ).

% psubset_imp_ex_mem
tff(fact_1906_uminus__unit__def,axiom,
    ! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).

% uminus_unit_def
tff(fact_1907_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M6: multiset(A),M9: multiset(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(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),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),M9)))) ).

% diff_size_le_size_Diff
tff(fact_1908_Collect__imp__eq,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_dd(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,bool),set(A),collect(A),P2))),aa(fun(A,bool),set(A),collect(A),Q2)) ).

% Collect_imp_eq
tff(fact_1909_subset__minus__empty,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_1910_subset__Compl__self__eq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),uminus_uminus(set(A)),A6)))
    <=> ( A6 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_1911_disjoint__alt__simp2,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) != A6 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp2
tff(fact_1912_disjoint__alt__simp1,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) = A6 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) ) ) ).

% disjoint_alt_simp1
tff(fact_1913_Int__Diff__disjoint,axiom,
    ! [A: $tType,A6: set(A),B5: 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)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_1914_Diff__triv,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) = A6 ) ) ).

% Diff_triv
tff(fact_1915_Diff__subset__conv,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),C4))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4))) ) ).

% Diff_subset_conv
tff(fact_1916_Diff__partition,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)) = B5 ) ) ).

% Diff_partition
tff(fact_1917_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_pos
tff(fact_1918_Un__Diff__Int,axiom,
    ! [A: $tType,A6: set(A),B5: 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)),minus_minus(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) = A6 ).

% Un_Diff_Int
tff(fact_1919_Int__Diff__Un,axiom,
    ! [A: $tType,A6: set(A),B5: 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)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = A6 ).

% Int_Diff_Un
tff(fact_1920_Diff__Int,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = 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)),minus_minus(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),C4)) ).

% Diff_Int
tff(fact_1921_Diff__Un,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) = 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)),minus_minus(set(A)),A6),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),C4)) ).

% Diff_Un
tff(fact_1922_Compl__Int,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).

% Compl_Int
tff(fact_1923_Compl__Un,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).

% Compl_Un
tff(fact_1924_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A3,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1925_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A3,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1926_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),uminus_uminus(set(A)),B5))) ) ).

% disjoint_eq_subset_Compl
tff(fact_1927_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_nonneg
tff(fact_1928_disjoint__alt__simp3,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),A6))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp3
tff(fact_1929_uminus__assn__def,axiom,
    ! [P2: assn] : aa(assn,assn,uminus_uminus(assn),P2) = abs_assn(aTP_Lamp_de(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),P2)) ).

% uminus_assn_def
tff(fact_1930_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),N)) ) ).

% pochhammer_rec
tff(fact_1931_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% pochhammer_Suc
tff(fact_1932_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z4: A,N: nat] : comm_s3205402744901411588hammer(A,Z4,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),aa(nat,A,semiring_1_of_nat(A),N))),comm_s3205402744901411588hammer(A,Z4,N)) ) ).

% pochhammer_rec'
tff(fact_1933_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N))
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_1934_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K2: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K2) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_1935_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K2) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_1936_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K2) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_1937_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z4: A,N: nat,M: nat] : comm_s3205402744901411588hammer(A,Z4,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z4,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),aa(nat,A,semiring_1_of_nat(A),N)),M)) ) ).

% pochhammer_product'
tff(fact_1938_normalize__denom__pos,axiom,
    ! [R3: product_prod(int,int),P3: int,Q5: int] :
      ( ( normalize(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q5) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q5)) ) ).

% normalize_denom_pos
tff(fact_1939_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z4: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( comm_s3205402744901411588hammer(A,Z4,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z4,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% pochhammer_product
tff(fact_1940_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R3),aa(nat,A,semiring_1_of_nat(A),K2))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R3),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),R3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R3)),one_one(A)),K2)) ) ).

% pochhammer_absorb_comp
tff(fact_1941_nonempty__has__size,axiom,
    ! [A: $tType,S: multiset(A)] :
      ( ( S != zero_zero(multiset(A)) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S))) ) ).

% nonempty_has_size
tff(fact_1942_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2)) ) ).

% pochhammer_minus
tff(fact_1943_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2)) ) ).

% pochhammer_minus'
tff(fact_1944_len_H__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & heap(B) )
     => ! [A3: array(B)] : array_len2(B,A,A3) = heap_Time_bind(nat,A,array_len(B,A3),aTP_Lamp_df(nat,heap_Time_Heap(A))) ) ).

% len'_def
tff(fact_1945_nat0__intermed__int__val,axiom,
    ! [N: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(nat,int,F3,N)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
              & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1946_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),From),To))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,aa(nat,fun(nat,nat),minus_minus(nat),To),From) ) ) ) ).

% slice_len
tff(fact_1947_nat__ivt__aux,axiom,
    ! [N: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(nat,int,F3,N)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
              & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1948_triangle__Suc,axiom,
    ! [N: nat] : nat_triangle(aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(N)),aa(nat,nat,suc,N)) ).

% triangle_Suc
tff(fact_1949_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
         => ( aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_1950_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_1951_DiffI,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),A6))
     => ( ~ pp(aa(set(A),bool,member(A,C3),B5))
       => pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5))) ) ) ).

% DiffI
tff(fact_1952_Diff__iff,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)))
    <=> ( pp(aa(set(A),bool,member(A,C3),A6))
        & ~ pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% Diff_iff
tff(fact_1953_Diff__idemp,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) ).

% Diff_idemp
tff(fact_1954_ComplI,axiom,
    ! [A: $tType,C3: A,A6: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,C3),A6))
     => pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),uminus_uminus(set(A)),A6))) ) ).

% ComplI
tff(fact_1955_Compl__iff,axiom,
    ! [A: $tType,C3: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),uminus_uminus(set(A)),A6)))
    <=> ~ pp(aa(set(A),bool,member(A,C3),A6)) ) ).

% Compl_iff
tff(fact_1956_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A6) = aa(set(A),set(A),uminus_uminus(set(A)),B5) )
    <=> ( A6 = B5 ) ) ).

% Compl_eq_Compl_iff
tff(fact_1957_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: 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),A3)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_1958_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_1959_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_1960_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% abs_le_self_iff
tff(fact_1961_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_1962_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_1963_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_1964_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_1965_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_1966_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B2))),zero_zero(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_1967_zabs__less__one__iff,axiom,
    ! [Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),Z4)),one_one(int)))
    <=> ( Z4 = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_1968_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),N)))
        <=> ( ( A3 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1969_DiffE,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)))
     => ~ ( pp(aa(set(A),bool,member(A,C3),A6))
         => pp(aa(set(A),bool,member(A,C3),B5)) ) ) ).

% DiffE
tff(fact_1970_ComplD,axiom,
    ! [A: $tType,C3: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),uminus_uminus(set(A)),A6)))
     => ~ pp(aa(set(A),bool,member(A,C3),A6)) ) ).

% ComplD
tff(fact_1971_DiffD1,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)))
     => pp(aa(set(A),bool,member(A,C3),A6)) ) ).

% DiffD1
tff(fact_1972_DiffD2,axiom,
    ! [A: $tType,C3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)))
     => ~ pp(aa(set(A),bool,member(A,C3),B5)) ) ).

% DiffD2
tff(fact_1973_double__complement,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)) = A6 ).

% double_complement
tff(fact_1974_uminus__set__def,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6))) ).

% uminus_set_def
tff(fact_1975_minus__set__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),B5))) ).

% minus_set_def
tff(fact_1976_Collect__neg__eq,axiom,
    ! [A: $tType,P2: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,bool),set(A),collect(A),P2)) ).

% Collect_neg_eq
tff(fact_1977_Compl__eq,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_dh(set(A),fun(A,bool),A6)) ).

% Compl_eq
tff(fact_1978_set__diff__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_di(set(A),fun(set(A),fun(A,bool)),A6),B5)) ).

% set_diff_eq
tff(fact_1979_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less_eq(multiset(A)),M6),N7))
        <=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
            | ( M6 = N7 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_1980_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
         => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),N7),M6)) ) ) ).

% mset_le_asym
tff(fact_1981_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K5: multiset(A),M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),K5),M6))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),K5),N7)) ) ) ) ).

% mset_le_trans
tff(fact_1982_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A)] : ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),M6)) ) ).

% mset_le_irrefl
tff(fact_1983_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
         => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),N7),M6)) ) ) ).

% mset_le_not_sym
tff(fact_1984_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A)] : ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),M6)) ) ).

% mset_le_not_refl
tff(fact_1985_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_self
tff(fact_1986_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% abs_le_D1
tff(fact_1987_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ).

% fact_mono_nat
tff(fact_1988_fact__ge__self,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),semiring_char_0_fact(nat,N))) ).

% fact_ge_self
tff(fact_1989_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_zero
tff(fact_1990_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_1991_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_1992_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq
tff(fact_1993_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))) ) ).

% abs_triangle_ineq2_sym
tff(fact_1994_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).

% abs_triangle_ineq3
tff(fact_1995_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).

% abs_triangle_ineq2
tff(fact_1996_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D3))) ) ) ) ).

% abs_mult_less
tff(fact_1997_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)) ) ) ) ).

% abs_leI
tff(fact_1998_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ).

% abs_le_D2
tff(fact_1999_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_2000_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_minus_self
tff(fact_2001_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_2002_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_2003_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K2)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc_Suc
tff(fact_2004_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_zero
tff(fact_2005_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K2) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_2006_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_2007_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_gt_zero
tff(fact_2008_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_1
tff(fact_2009_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ).

% fact_mono
tff(fact_2010_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2)),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_pochhammer'
tff(fact_2011_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K2)) ) ).

% gbinomial_addition_formula
tff(fact_2012_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K2)) = 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),K2)),aa(nat,A,gbinomial(A,A3),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1
tff(fact_2013_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),A3) = 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),K2)),aa(nat,A,gbinomial(A,A3),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1'
tff(fact_2014_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E2)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2015_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2016_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
              | pp(aa(A,bool,aa(A,fun(A,bool),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),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2017_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K2)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,gbinomial(A,A3),K2))) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_2018_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
            & ( ( A3 = B2 )
              | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2019_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & ( ( B2 = A3 )
              | ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2020_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A))) ) ).

% abs_minus_le_zero
tff(fact_2021_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq4
tff(fact_2022_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A,C3: A,D3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))))) ) ).

% abs_diff_triangle_ineq
tff(fact_2023_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R3))) ) ) ) ).

% abs_diff_le_iff
tff(fact_2024_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ) ).

% abs_if
tff(fact_2025_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X5: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X5) = aa(A,A,uminus_uminus(A),X5) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X5) = X5 ) ) ) ) ).

% abs_if_raw
tff(fact_2026_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_2027_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R3))) ) ) ) ).

% abs_diff_less_iff
tff(fact_2028_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% abs_div_pos
tff(fact_2029_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),N))) ) ).

% zero_le_power_abs
tff(fact_2030_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_2031_zabs__def,axiom,
    ! [I2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I2) = aa(int,int,uminus_uminus(int),I2) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I2) = I2 ) ) ) ).

% zabs_def
tff(fact_2032_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_2033_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero(int) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K2,L))),aa(int,int,abs_abs(int),L))) ) ).

% abs_mod_less
tff(fact_2034_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_2035_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,N),N)))) ) ).

% fact_le_power
tff(fact_2036_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K2)) ) ).

% Suc_times_gbinomial
tff(fact_2037_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,M: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K2))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2038_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),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),X)))) ) ).

% abs_add_one_gt_zero
tff(fact_2039_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] :
          ( ( ( K2 = zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K2) = one_one(A) ) )
          & ( ( K2 != zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_dj(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K2)) ) ) ) ) ).

% gbinomial_code
tff(fact_2040_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K2))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))) ).

% nat_abs_triangle_ineq
tff(fact_2041_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),R3)))),aa(nat,nat,power_power(nat,N),R3))) ) ).

% fact_div_fact_le_pow
tff(fact_2042_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2)))),aa(nat,A,gbinomial(A,A3),K2)) ) ).

% gbinomial_factors
tff(fact_2043_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_rec
tff(fact_2044_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A))),K2)) ) ).

% gbinomial_minus
tff(fact_2045_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
         => ( aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K2)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2046_nat__abs__int__diff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A3) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_2047_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N)) )
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,M)),K2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(nat,int,F3,N)))
           => ? [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I3))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
                & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2048_decr__lemma,axiom,
    ! [D3: int,X: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),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,aa(int,fun(int,int),minus_minus(int),X),Z4))),one_one(int))),D3))),Z4)) ) ).

% decr_lemma
tff(fact_2049_incr__lemma,axiom,
    ! [D3: int,Z4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),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,aa(int,fun(int,int),minus_minus(int),X),Z4))),one_one(int))),D3)))) ) ).

% incr_lemma
tff(fact_2050_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) ) )
        | ? [L2: int,K3: int,Q7: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),zero_zero(int)) )
            & ( L2 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q7),L2) ) )
        | ? [R4: int,L2: int,K3: int,Q7: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),R4) )
            & ( aa(int,int,sgn_sgn(int),R4) = aa(int,int,sgn_sgn(int),L2) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R4)),aa(int,int,abs_abs(int),L2)))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q7),L2)),R4) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2051_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) ) )
       => ( ! [Q4: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A22) ) ) )
         => ~ ! [R: int,Q4: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R) )
               => ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),A22) )
                 => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R)),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),Q4),A22)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2052_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2053_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From)))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I2)) ) ) ) ) ).

% slice_nth
tff(fact_2054_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K2: int,Q5: int] :
      ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),L) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),L)))
       => ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L)),R3) )
         => eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2055_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2056_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,sgn_sgn(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_2057_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_greater
tff(fact_2058_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_2059_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_2060_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_2061_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Ys2),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2062_Skolem__list__nth,axiom,
    ! [A: $tType,K2: nat,P2: fun(nat,fun(A,bool))] :
      ( ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K2))
         => ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P2,I),X_12)) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K2 )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K2))
             => pp(aa(A,bool,aa(nat,fun(A,bool),P2,I),aa(nat,A,nth(A,Xs3),I))) ) ) ) ).

% Skolem_list_nth
tff(fact_2063_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2064_obtain__list__from__elements,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(nat,bool))] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => ? [Li: A] : pp(aa(nat,bool,aa(A,fun(nat,bool),P2,Li),I3)) )
     => ~ ! [L3: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L3) = N )
           => ~ ! [I4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
                 => pp(aa(nat,bool,aa(A,fun(nat,bool),P2,aa(nat,A,nth(A,L3),I4)),I4)) ) ) ) ).

% obtain_list_from_elements
tff(fact_2065_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_2066_nth__rotate1,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_2067_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_1_pos
tff(fact_2068_zsgn__def,axiom,
    ! [I2: int] :
      ( ( ( I2 = zero_zero(int) )
       => ( aa(int,int,sgn_sgn(int),I2) = zero_zero(int) ) )
      & ( ( I2 != zero_zero(int) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
           => ( aa(int,int,sgn_sgn(int),I2) = one_one(int) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
           => ( aa(int,int,sgn_sgn(int),I2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).

% zsgn_def
tff(fact_2069_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_2070_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = one_one(A) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).

% sgn_if
tff(fact_2071_nth__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,Xs: list(A),A3: array(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
         => hoare_hoare_triple(A,snga_assn(A,A3,Xs),array_nth(A,A3,I2),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_dk(nat,fun(list(A),fun(array(A),fun(A,assn))),I2),Xs),A3)) ) ) ).

% nth_rule
tff(fact_2072_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys2)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(list(B),nat,size_size(list(B)),Ys2)))),aa(nat,B,nth(B,Ys2),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys2)))) ) ) ).

% product_nth
tff(fact_2073_sorted__in__between,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: nat,J2: nat,L: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),I2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L)))
             => ( sorted_wrt(A,ord_less_eq(A),L)
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I2)),X))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,nth(A,L),J2)))
                   => ~ ! [K: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),J2))
                           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),K)),X))
                             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_2074_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2075_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_2076_nth__step__trancl,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),N: nat,M: nat] :
      ( ! [N4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))))
         => pp(aa(set(product_prod(A,A)),bool,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,N4))),aa(nat,A,nth(A,Xs),N4))),R2)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => pp(aa(set(product_prod(A,A)),bool,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),N)),aa(nat,A,nth(A,Xs),M))),transitive_trancl(A,R2))) ) ) ) ).

% nth_step_trancl
tff(fact_2077_nth__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A),M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,M),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_2078_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_2079_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_dl(A,fun(A,bool)),Xs) ).

% sorted_wrt_true
tff(fact_2080_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R3)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),R3))
       => ~ ! [B4: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4)),transitive_trancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22)),R3)) ) ) ) ).

% trancl.cases
tff(fact_2081_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R3)))
    <=> ( ? [A9: A,B7: A] :
            ( ( A1 = A9 )
            & ( A22 = B7 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),B7)),R3)) )
        | ? [A9: A,B7: A,C5: A] :
            ( ( A1 = A9 )
            & ( A22 = C5 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),B7)),transitive_trancl(A,R3)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5)),R3)) ) ) ) ).

% trancl.simps
tff(fact_2082_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3))) ) ).

% trancl.r_into_trancl
tff(fact_2083_tranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ~ ! [C2: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C2),B2)),R3)) ) ) ) ).

% tranclE
tff(fact_2084_trancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Z4: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),transitive_trancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),transitive_trancl(A,R3))) ) ) ).

% trancl_trans
tff(fact_2085_trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),R3))
           => pp(aa(A,bool,P2,Y3)) )
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_trancl(A,R3)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
               => ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,P2,Z3)) ) ) )
         => pp(aa(A,bool,P2,B2)) ) ) ) ).

% trancl_induct
tff(fact_2086_r__r__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R2))) ) ) ).

% r_r_into_trancl
tff(fact_2087_converse__tranclE,axiom,
    ! [A: $tType,X: A,Z4: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),transitive_trancl(A,R3)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),R3))
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R3))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),transitive_trancl(A,R3))) ) ) ) ).

% converse_tranclE
tff(fact_2088_irrefl__trancl__rD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_2089_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R3))) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_2090_trancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_trancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R3))) ) ) ).

% trancl_into_trancl2
tff(fact_2091_trancl__trans__induct,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),P2: fun(A,fun(A,bool))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R3)))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Y3)) )
       => ( ! [X3: A,Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),transitive_trancl(A,R3)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Y3))
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),transitive_trancl(A,R3)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),P2,Y3),Z3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Z3)) ) ) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y)) ) ) ) ).

% trancl_trans_induct
tff(fact_2092_converse__trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2)),R3))
           => pp(aa(A,bool,P2,Y3)) )
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_trancl(A,R3)))
               => ( pp(aa(A,bool,P2,Z3))
                 => pp(aa(A,bool,P2,Y3)) ) ) )
         => pp(aa(A,bool,P2,A3)) ) ) ) ).

% converse_trancl_induct
tff(fact_2093_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_trancl(product_prod(A,B),R3)))
     => ( ! [A5: A,B4: B] :
            ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),R3))
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),transitive_trancl(product_prod(A,B),R3)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R3))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_2094_trancl__mono,axiom,
    ! [A: $tType,P3: product_prod(A,A),R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_trancl(A,S2))) ) ) ).

% trancl_mono
tff(fact_2095_trancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V: set(product_prod(A,A)),X: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U2),V))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_trancl(A,U2)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_trancl(A,V))) ) ) ).

% trancl_mono_mp
tff(fact_2096_trancl__sub,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),transitive_trancl(A,R2))) ).

% trancl_sub
tff(fact_2097_rotate__rotate,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,M),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),Xs) ).

% rotate_rotate
tff(fact_2098_rotate__add,axiom,
    ! [A: $tType,M: nat,N: nat] : rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),rotate(A,M)),rotate(A,N)) ).

% rotate_add
tff(fact_2099_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_2100_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I2: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Ns)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,nth(nat,Ns),I2))) ) ) ).

% sorted_wrt_less_idx
tff(fact_2101_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P2,Xs)
    <=> ! [I: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_2102_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),Xs: list(A),I2: nat,J2: nat] :
      ( sorted_wrt(A,P2,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_2103_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,fun(A,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => sorted_wrt(A,P2,Xs) ) ).

% sorted_wrt01
tff(fact_2104_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_2105_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_2106_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I)))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_2107_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_2108_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ) ).

% sorted_nth_mono
tff(fact_2109_nth__zip,axiom,
    ! [A: $tType,B: $tType,I2: nat,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys2)))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys2)),I2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys2),I2)) ) ) ) ).

% nth_zip
tff(fact_2110_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_2111_same__fst__trancl,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),R2: fun(A,set(product_prod(B,B)))] : transitive_trancl(product_prod(A,B),same_fst(A,B,P2,R2)) = same_fst(A,B,P2,aTP_Lamp_dm(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R2)) ).

% same_fst_trancl
tff(fact_2112_find__Some__iff,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),X) )
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I)))
          & ( X = aa(nat,A,nth(A,Xs),I) )
          & ! [J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I))
             => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% find_Some_iff
tff(fact_2113_find__Some__iff2,axiom,
    ! [A: $tType,X: A,P2: fun(A,bool),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),X) = find(A,P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I)))
          & ( X = aa(nat,A,nth(A,Xs),I) )
          & ! [J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I))
             => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% find_Some_iff2
tff(fact_2114_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2115_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2116_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% numeral_le_iff
tff(fact_2117_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% numeral_less_iff
tff(fact_2118_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% numeral_plus_numeral
tff(fact_2119_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V2: num,W2: num,Z4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Z4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),W2))),Z4) ) ).

% add_numeral_left
tff(fact_2120_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).

% power_add_numeral2
tff(fact_2121_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% power_add_numeral
tff(fact_2122_zip__eq__zip__same__len,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),A4: list(A),B3: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
       => ( ( zip(A,B,A3,B2) = zip(A,B,A4,B3) )
        <=> ( ( A3 = A4 )
            & ( B2 = B3 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_2123_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).

% neg_numeral_le_iff
tff(fact_2124_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = 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),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C3)) ) ).

% distrib_left_numeral
tff(fact_2125_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B2: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).

% distrib_right_numeral
tff(fact_2126_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).

% neg_numeral_less_iff
tff(fact_2127_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_2128_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),W2)))),Y) ) ).

% semiring_norm(168)
tff(fact_2129_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% diff_numeral_simps(3)
tff(fact_2130_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% diff_numeral_simps(2)
tff(fact_2131_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys2: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),X),Y)),zip(A,B,Xs,Ys2)) ).

% zip_Cons_Cons
tff(fact_2132_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2133_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2134_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2135_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2136_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2137_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2138_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2139_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2140_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,A3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2141_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A3)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2142_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,A3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2143_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A3)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2144_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I2: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N))) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2145_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N)),X)) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2146_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I2: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N))) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2147_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N)),X)) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2148_nat__numeral__as__int,axiom,
    ! [X5: num] : aa(num,nat,numeral_numeral(nat),X5) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X5)) ).

% nat_numeral_as_int
tff(fact_2149_int__ops_I3_J,axiom,
    ! [N: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),N)) = aa(num,int,numeral_numeral(int),N) ).

% int_ops(3)
tff(fact_2150_list__tail__coinc,axiom,
    ! [A: $tType,N12: A,R12: list(A),N23: A,R23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N12),R12) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N23),R23) )
     => ( ( N12 = N23 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
tff(fact_2151_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( zip(A,B,Xs,Ys2) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
     => ~ ! [X3: A,Xs4: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
           => ! [Y3: B,Ys3: list(B)] :
                ( ( Ys2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
                 => ( Xys != zip(A,B,Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_2152_find_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P2,X))
       => ( find(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,option(A),some(A),X) ) )
      & ( ~ pp(aa(A,bool,P2,X))
       => ( find(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = find(A,P2,Xs) ) ) ) ).

% find.simps(2)
tff(fact_2153_pair__list__split,axiom,
    ! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
      ~ ! [L1: list(A),L22: list(B)] :
          ( ( L = zip(A,B,L1,L22) )
         => ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L22) )
           => ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L22) ) ) ) ).

% pair_list_split
tff(fact_2154_zip__inj,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),A4: list(A),B3: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B3) )
       => ( ( zip(A,B,A3,B2) = zip(A,B,A4,B3) )
         => ( ( A3 = A4 )
            & ( B2 = B3 ) ) ) ) ) ).

% zip_inj
tff(fact_2155_find__SomeD_I1_J,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),X) )
     => pp(aa(A,bool,P2,X)) ) ).

% find_SomeD(1)
tff(fact_2156_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_2157_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_le_numeral
tff(fact_2158_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_2159_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_less_numeral
tff(fact_2160_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% one_le_numeral
tff(fact_2161_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_2162_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2163_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_le_numeral
tff(fact_2164_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_2165_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2166_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_less_numeral
tff(fact_2167_length__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_2168_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2) ) ) ).

% impossible_Cons
tff(fact_2169_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_2170_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_le_neg_numeral
tff(fact_2171_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_2172_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_less_neg_numeral
tff(fact_2173_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_2174_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_2175_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_le_numeral
tff(fact_2176_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_2177_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_2178_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_le_neg_numeral
tff(fact_2179_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_2180_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_less_numeral
tff(fact_2181_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_2182_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_less_neg_numeral
tff(fact_2183_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2184_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ? [X4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_2185_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2186_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2187_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_2188_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2189_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2190_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2191_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2192_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_2193_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,X: A,Xs: list(A)] :
      ( ( ( ( Begin = zero_zero(nat) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
       => ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),slice(A,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero(nat) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
       => ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Begin),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs) ) ) ) ).

% slice_Cons
tff(fact_2194_same__fstI,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),X: A,Y7: B,Y: B,R2: fun(A,set(product_prod(B,B)))] :
      ( pp(aa(A,bool,P2,X))
     => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y)),aa(A,set(product_prod(B,B)),R2,X)))
       => pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y))),same_fst(A,B,P2,R2))) ) ) ).

% same_fstI
tff(fact_2195_upto__aux__rec,axiom,
    ! [J2: int,I2: int,Js: list(int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I2))
       => ( upto_aux(I2,J2,Js) = Js ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I2))
       => ( upto_aux(I2,J2,Js) = upto_aux(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J2),Js)) ) ) ) ).

% upto_aux_rec
tff(fact_2196_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2197_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2198_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2199_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2200_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2201_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z4)) ) ) ).

% of_int_le_iff
tff(fact_2202_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z4)) ) ) ).

% of_int_less_iff
tff(fact_2203_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: int,Z4: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),Z4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z4)) ) ).

% of_int_add
tff(fact_2204_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z4: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z4))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z4) ) ).

% ceiling_add_of_int
tff(fact_2205_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z4)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_2206_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4)) ) ) ).

% of_int_0_le_iff
tff(fact_2207_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z4)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_2208_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z4)) ) ) ).

% of_int_0_less_iff
tff(fact_2209_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z4)) ) ) ).

% of_int_numeral_le_iff
tff(fact_2210_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z4)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_le_numeral_iff
tff(fact_2211_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),N)),Z4)) ) ) ).

% of_int_numeral_less_iff
tff(fact_2212_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z4)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_less_numeral_iff
tff(fact_2213_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_2214_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z4)) ) ) ).

% of_int_1_le_iff
tff(fact_2215_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),one_one(int))) ) ) ).

% of_int_less_1_iff
tff(fact_2216_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z4)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z4)) ) ) ).

% of_int_1_less_iff
tff(fact_2217_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_le_zero
tff(fact_2218_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% zero_less_ceiling
tff(fact_2219_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_numeral
tff(fact_2220_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_less_one
tff(fact_2221_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% one_le_ceiling
tff(fact_2222_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_2223_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).

% ceiling_le_one
tff(fact_2224_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).

% one_less_ceiling
tff(fact_2225_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_add_numeral
tff(fact_2226_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_2227_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z4)) = aa(int,A,ring_1_of_int(A),Z4) ) ) ) ).

% of_nat_nat
tff(fact_2228_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,B2),W2)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2229_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,power_power(int,B2),W2))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2230_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,B2),W2)),X)) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2231_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,power_power(int,B2),W2))) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2232_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).

% ceiling_less_zero
tff(fact_2233_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).

% zero_le_ceiling
tff(fact_2234_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_2235_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_2236_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2237_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2238_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2239_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2240_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2241_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2242_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2243_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ).

% le_of_int_ceiling
tff(fact_2244_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),Z4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z4))) ) ) ).

% ceiling_le_iff
tff(fact_2245_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z4)),X)) ) ) ).

% less_ceiling_iff
tff(fact_2246_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_le_of_int
tff(fact_2247_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),X)) ) ).

% ex_of_int_less
tff(fact_2248_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).

% ex_less_of_int
tff(fact_2249_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).

% ceiling_correct
tff(fact_2250_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z4)))
           => ( archimedean_ceiling(A,X) = Z4 ) ) ) ) ).

% ceiling_unique
tff(fact_2251_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archimedean_ceiling(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3))) ) ) ) ).

% ceiling_eq_iff
tff(fact_2252_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,bool),T5: A] :
          ( pp(aa(int,bool,P2,archimedean_ceiling(A,T5)))
        <=> ! [I: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I)),one_one(A))),T5))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T5),aa(int,A,ring_1_of_int(A),I))) )
             => pp(aa(int,bool,P2,I)) ) ) ) ).

% ceiling_split
tff(fact_2253_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A)))) ) ) ).

% ceiling_less_iff
tff(fact_2254_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A))),X)) ) ) ).

% le_ceiling_iff
tff(fact_2255_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_2256_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% ceiling_less_cancel
tff(fact_2257_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),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),Q5)))),Q5))) ) ) ).

% ceiling_divide_upper
tff(fact_2258_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(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),Q5)))),one_one(A))),Q5)),P3)) ) ) ).

% ceiling_divide_lower
tff(fact_2259_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R3),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R3))))) ) ).

% of_nat_ceiling
tff(fact_2260_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))) ) ).

% ceiling_add_le
tff(fact_2261_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z4))) ) ) ).

% of_int_nonneg
tff(fact_2262_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z4))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z4))) ) ) ).

% of_int_pos
tff(fact_2263_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_2264_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% of_int_lessD
tff(fact_2265_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),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_2266_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int)))))
          & ! [Y5: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y5)),X))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y5),one_one(int))))) )
             => ( Y5 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_2267_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),X)) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2268_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le
tff(fact_2269_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: int] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K2) = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K2)))) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
           => ( aa(int,A,ring_1_of_int(A),K2) = aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K2)) ) ) ) ) ).

% of_int_of_nat
tff(fact_2270_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_2271_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_2272_shift__def,axiom,
    ! [B: $tType,A: $tType,Lab: fun(list(A),B),K2: A,X5: list(A)] : bNF_Greatest_shift(A,B,Lab,K2,X5) = aa(list(A),B,Lab,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K2),X5)) ).

% shift_def
tff(fact_2273_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R3: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),lex_prod(A,B,R3,S2)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A4)),R3))
        | ( ( A3 = A4 )
          & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B3)),S2)) ) ) ) ).

% in_lex_prod
tff(fact_2274_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,member(B,A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A3)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_2275_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)))) ) ) ).

% floor_less_neg_numeral
tff(fact_2276_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_2277_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
            | pp(aa(set(A),bool,member(A,Y),ring_1_Ints(A))) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_2278_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ) ).

% zero_le_floor
tff(fact_2279_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A))) ) ) ).

% floor_less_zero
tff(fact_2280_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),X)) ) ) ).

% numeral_le_floor
tff(fact_2281_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% zero_less_floor
tff(fact_2282_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_le_zero
tff(fact_2283_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_numeral
tff(fact_2284_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% one_le_floor
tff(fact_2285_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_less_one
tff(fact_2286_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_2287_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_2288_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,member(A,B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_add
tff(fact_2289_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))) ) ) ).

% floor_mono
tff(fact_2290_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% floor_less_cancel
tff(fact_2291_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)) ) ).

% of_int_floor_le
tff(fact_2292_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X))) ) ).

% floor_le_ceiling
tff(fact_2293_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2294_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,member(B,A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A3)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_2295_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z4)),X)) ) ) ).

% le_floor_iff
tff(fact_2296_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)))) ) ).

% le_floor_add
tff(fact_2297_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),Z4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z4))) ) ) ).

% floor_less_iff
tff(fact_2298_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z4: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z4) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z4))) ) ).

% floor_add_int
tff(fact_2299_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z4),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z4)),X)) ) ).

% int_add_floor
tff(fact_2300_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_2301_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R3)))),R3)) ) ) ).

% of_nat_floor
tff(fact_2302_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).

% one_add_floor
tff(fact_2303_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).

% ceiling_diff_floor_le_1
tff(fact_2304_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = archim6421214686448440834_floor(A,X) ) )
          & ( ( X != aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ) ) ) ).

% ceiling_altdef
tff(fact_2305_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_2306_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
         => ( ( X != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_2307_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_2308_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,member(A,Y),ring_1_Ints(A)))
           => ( ( X = Y )
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_2309_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z4)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z4 ) ) ) ) ).

% floor_unique
tff(fact_2310_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_2311_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,bool),T5: A] :
          ( pp(aa(int,bool,P2,archim6421214686448440834_floor(A,T5)))
        <=> ! [I: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I)),T5))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T5),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I)),one_one(A)))) )
             => pp(aa(int,bool,P2,I)) ) ) ) ).

% floor_split
tff(fact_2312_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ).

% le_mult_floor
tff(fact_2313_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_2314_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z4))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z4)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_2315_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).

% floor_correct
tff(fact_2316_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),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),Q5)))),Q5)),P3)) ) ) ).

% floor_divide_lower
tff(fact_2317_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q5))
         => pp(aa(A,bool,aa(A,fun(A,bool),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),Q5)))),one_one(A))),Q5))) ) ) ).

% floor_divide_upper
tff(fact_2318_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_2319_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: A] :
          ( ( archimedean_frac(A,X) = A3 )
        <=> ( pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_2320_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one))) ) ).

% diff_numeral_special(4)
tff(fact_2321_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one),N)) ) ).

% diff_numeral_special(3)
tff(fact_2322_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_2323_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_2324_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),one),N)) ).

% semiring_norm(68)
tff(fact_2325_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),one)) ).

% semiring_norm(75)
tff(fact_2326_Suc__numeral,axiom,
    ! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one)) ).

% Suc_numeral
tff(fact_2327_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),archimedean_frac(A,X)))
        <=> ~ pp(aa(set(A),bool,member(A,X),ring_1_Ints(A))) ) ) ).

% frac_gt_0_iff
tff(fact_2328_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))))
        <=> ( M != one ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2329_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M != one ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2330_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one),N)) ) ).

% one_plus_numeral
tff(fact_2331_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one)) ) ).

% numeral_plus_one
tff(fact_2332_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one)) ) ) ).

% numeral_le_one_iff
tff(fact_2333_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one),N)) ) ) ).

% one_less_numeral_iff
tff(fact_2334_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one) = inc(X) ).

% add_One
tff(fact_2335_add__One__commute,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one) ).

% add_One_commute
tff(fact_2336_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),X),one))
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_2337_add__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) ).

% add_inc
tff(fact_2338_mult__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)),X) ).

% mult_inc
tff(fact_2339_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X))) ) ).

% frac_ge_0
tff(fact_2340_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).

% frac_lt_1
tff(fact_2341_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_2342_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_2343_Suc__nat__number__of__add,axiom,
    ! [V2: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),one))),N) ).

% Suc_nat_number_of_add
tff(fact_2344_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% frac_eq
tff(fact_2345_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_2346_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z4: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),Z4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))),comm_s3205402744901411588hammer(A,Z4,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),N)) ) ).

% pochhammer_double
tff(fact_2347_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q5: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),zero_zero(int)))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),R3)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_2348_execute__change,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [F3: fun(A,A),R3: ref(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,ref_change(A,F3,R3)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(A,A,F3,ref_get(A,H,R3))),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,R3,aa(A,A,F3,ref_get(A,H,R3)),H)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one))))) ) ).

% execute_change
tff(fact_2349_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),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),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2350_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q5: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),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),aa(num,num,bit0,one))),R3)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2351_neg__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),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),aa(num,num,bit0,one))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),A3)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A3))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2352_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa(num,num,bit0,X2) = aa(num,num,bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_2353_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_2354_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(6)
tff(fact_2355_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(78)
tff(fact_2356_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(71)
tff(fact_2357_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(80)
tff(fact_2358_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(73)
tff(fact_2359_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one),one) = aa(num,num,bit0,one) ).

% semiring_norm(2)
tff(fact_2360_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(9)
tff(fact_2361_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(7)
tff(fact_2362_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),one)) ).

% semiring_norm(69)
tff(fact_2363_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one),aa(num,num,bit0,N))) ).

% semiring_norm(76)
tff(fact_2364_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(72)
tff(fact_2365_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(81)
tff(fact_2366_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),one)) ).

% semiring_norm(70)
tff(fact_2367_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one),aa(num,num,bit1,N))) ).

% semiring_norm(77)
tff(fact_2368_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),one)) ).

% semiring_norm(10)
tff(fact_2369_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one)) ).

% semiring_norm(8)
tff(fact_2370_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),one) = aa(num,num,bit1,M) ).

% semiring_norm(5)
tff(fact_2371_semiring__norm_I4_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one)) ).

% semiring_norm(4)
tff(fact_2372_semiring__norm_I3_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one),aa(num,num,bit0,N)) = aa(num,num,bit1,N) ).

% semiring_norm(3)
tff(fact_2373_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)))) ).

% semiring_norm(16)
tff(fact_2374_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(79)
tff(fact_2375_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(74)
tff(fact_2376_one__add__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)) ) ) ).

% one_add_one
tff(fact_2377_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_2378_add__2__eq__Suc,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_2379_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) = M ).

% add_self_div_2
tff(fact_2380_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_2381_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_2382_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2383_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_2384_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),aa(num,num,bit0,one))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2385_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = 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,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_2386_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V2: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V2)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one))),M),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2387_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_2388_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V2: 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),V2)) = 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,one))),M)),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2389_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),X)) ) ) ).

% one_less_floor
tff(fact_2390_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))) ) ) ).

% floor_le_one
tff(fact_2391_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2392_zmod__numeral__Bit1,axiom,
    ! [V2: num,W2: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V2)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W2))) = 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),aa(num,num,bit0,one))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V2),aa(num,int,numeral_numeral(int),W2)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_2393_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa(num,num,bit0,X2) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_2394_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_2395_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa(num,num,bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_2396_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_Bit0
tff(fact_2397_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_code(2)
tff(fact_2398_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_2399_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_2400_num_Osize_I5_J,axiom,
    ! [X2: num] : aa(num,nat,size_size(num),aa(num,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_2401_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_code(3)
tff(fact_2402_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z4: A,W2: num] : aa(nat,A,power_power(A,Z4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),aa(nat,A,power_power(A,Z4),aa(num,nat,numeral_numeral(nat),W2)))),aa(nat,A,power_power(A,Z4),aa(num,nat,numeral_numeral(nat),W2))) ) ).

% power_numeral_odd
tff(fact_2403_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z4: A,W2: num] : aa(nat,A,power_power(A,Z4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Z4),aa(num,nat,numeral_numeral(nat),W2))),aa(nat,A,power_power(A,Z4),aa(num,nat,numeral_numeral(nat),W2))) ) ).

% power_numeral_even
tff(fact_2404_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_2405_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),Z4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),Z4) ) ).

% mult_2
tff(fact_2406_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z4),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),Z4) ) ).

% mult_2_right
tff(fact_2407_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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),aa(num,num,bit0,one))),A3)),B2) ) ).

% left_add_twice
tff(fact_2408_nat__1__add__1,axiom,
    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)) ).

% nat_1_add_1
tff(fact_2409_less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))) ).

% less_exp
tff(fact_2410_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_2411_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,nat,power_power(nat,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_2412_self__le__ge2__pow,axiom,
    ! [K2: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,K2),M))) ) ).

% self_le_ge2_pow
tff(fact_2413_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% half_gt_zero_iff
tff(fact_2414_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))) ) ) ).

% half_gt_zero
tff(fact_2415_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ).

% zero_le_power2
tff(fact_2416_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2417_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% power2_le_imp_le
tff(fact_2418_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_2419_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one))),M),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2420_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_2421_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2422_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2423_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2424_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ) ).

% abs_le_square_iff
tff(fact_2425_nat__induct2,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( pp(aa(nat,bool,P2,one_one(nat)))
       => ( ! [N4: nat] :
              ( pp(aa(nat,bool,P2,N4))
             => pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_induct2
tff(fact_2426_diff__le__diff__pow,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,K2),M)),aa(nat,nat,power_power(nat,K2),N)))) ) ).

% diff_le_diff_pow
tff(fact_2427_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2))),B2))
           => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2)) = modulo_modulo(A,A3,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2428_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% power2_less_imp_less
tff(fact_2429_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2430_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))) ) ).

% sum_power2_ge_zero
tff(fact_2431_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2432_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_2433_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) = 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,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),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),aa(num,num,bit0,one))),X)),Y)) ) ).

% power2_sum
tff(fact_2434_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2435_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))) ) ).

% zero_le_even_power'
tff(fact_2436_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2437_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2438_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_less_1
tff(fact_2439_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2440_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_2441_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2442_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),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),aa(num,num,bit0,one))),X)),Y)) ) ).

% power2_diff
tff(fact_2443_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2444_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_2445_effect__changeE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [F3: fun(A,A),R6: ref(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
          ( heap_Time_effect(A,ref_change(A,F3,R6),H,H5,R3,N)
         => ~ ( ( H5 = ref_set(A,R6,aa(A,A,F3,ref_get(A,H,R6)),H) )
             => ( ( R3 = aa(A,A,F3,ref_get(A,H,R6)) )
               => ( N != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one)) ) ) ) ) ) ).

% effect_changeE
tff(fact_2446_effect__changeI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H5: heap_ext(product_unit),R3: ref(A),F3: fun(A,A),H: heap_ext(product_unit),X: A,N: nat] :
          ( ( H5 = ref_set(A,R3,aa(A,A,F3,ref_get(A,H,R3)),H) )
         => ( ( X = aa(A,A,F3,ref_get(A,H,R3)) )
           => ( ( N = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one)) )
             => heap_Time_effect(A,ref_change(A,F3,R3),H,H5,X,N) ) ) ) ) ).

% effect_changeI
tff(fact_2447_ex__power__ivl1,axiom,
    ! [B2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K2))
       => ? [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),N4)),K2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_2448_ex__power__ivl2,axiom,
    ! [B2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2))
       => ? [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,B2),N4)),K2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_2449_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2450_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2))),B2) = modulo_modulo(A,A3,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2451_pos__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_2452_neg__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),A3)) = 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))),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_2453_pos__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( 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),aa(num,num,bit0,one))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),modulo_modulo(int,B2,A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_2454_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% half_negative_int_iff
tff(fact_2455_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% half_nonnegative_int_iff
tff(fact_2456_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R3: A,Q5: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),R3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),Q5)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R3),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),R3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),Q5)),R3) ) ) ) ) ).

% divmod_step_eq
tff(fact_2457_int__bit__induct,axiom,
    ! [P2: fun(int,bool),K2: int] :
      ( pp(aa(int,bool,P2,zero_zero(int)))
     => ( pp(aa(int,bool,P2,aa(int,int,uminus_uminus(int),one_one(int))))
       => ( ! [K: int] :
              ( pp(aa(int,bool,P2,K))
             => ( ( K != zero_zero(int) )
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))))) ) )
         => ( ! [K: int] :
                ( pp(aa(int,bool,P2,K))
               => ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => pp(aa(int,bool,P2,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),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) )
           => pp(aa(int,bool,P2,K2)) ) ) ) ) ).

% int_bit_induct
tff(fact_2458_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2459_nat__bit__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N4: nat] :
            ( pp(aa(nat,bool,P2,N4))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
             => pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N4))) ) )
       => ( ! [N4: nat] :
              ( pp(aa(nat,bool,P2,N4))
             => pp(aa(nat,bool,P2,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N4)))) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_bit_induct
tff(fact_2460_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2461_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2462_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ).

% div_exp_eq
tff(fact_2463_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),zero_zero(int))) ).

% not_exp_less_eq_0_int
tff(fact_2464_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = A3 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2465_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat,M: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A3,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_2466_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))) ) ).

% set_bit_0
tff(fact_2467_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),bit_se2638667681897837118et_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))) ) ).

% unset_bit_Suc
tff(fact_2468_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),bit_se8732182000553998342ip_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))) ) ).

% flip_bit_Suc
tff(fact_2469_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),bit_se5668285175392031749et_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))) ) ).

% set_bit_Suc
tff(fact_2470_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_2471_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))),aa(int,A,ring_1_of_int(A),Y)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_2472_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2638667681897837118et_bit(int,N,K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2473_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5668285175392031749et_bit(int,N,K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2474_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,N,K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2475_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2638667681897837118et_bit(int,N,K2)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% unset_bit_negative_int_iff
tff(fact_2476_set__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5668285175392031749et_bit(int,N,K2)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% set_bit_negative_int_iff
tff(fact_2477_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se8732182000553998342ip_bit(int,N,K2)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% flip_bit_negative_int_iff
tff(fact_2478_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K2))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_2479_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_2480_signed__take__bit__add,axiom,
    ! [N: nat,K2: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)) ).

% signed_take_bit_add
tff(fact_2481_unset__bit__less__eq,axiom,
    ! [N: nat,K2: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se2638667681897837118et_bit(int,N,K2)),K2)) ).

% unset_bit_less_eq
tff(fact_2482_set__bit__greater__eq,axiom,
    ! [K2: int,N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),bit_se5668285175392031749et_bit(int,N,K2))) ).

% set_bit_greater_eq
tff(fact_2483_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y))) ) ) ).

% round_mono
tff(fact_2484_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X))) ) ).

% floor_le_round
tff(fact_2485_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X))) ) ).

% ceiling_ge_round
tff(fact_2486_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_2487_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z4: A,M: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z4))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),aa(int,A,ring_1_of_int(A),M))))) ) ).

% round_diff_minimal
tff(fact_2488_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2489_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),K2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),K2)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2490_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K2: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2491_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),K2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))),K2)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2492_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2493_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),K2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(nat,nat,suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2494_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2) = K2 )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))),K2))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2495_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2496_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2497_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))) ) ).

% round_def
tff(fact_2498_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))) ) ).

% signed_take_bit_Suc
tff(fact_2499_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))) ) ).

% of_int_round_le
tff(fact_2500_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_2501_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_2502_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))) ) ).

% of_int_round_abs_le
tff(fact_2503_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_2504_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_2505_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_2506_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_2507_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2508_concat__bit__Suc,axiom,
    ! [N: nat,K2: int,L: int] : bit_concat_bit(aa(nat,nat,suc,N),K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_concat_bit(N,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L))) ).

% concat_bit_Suc
tff(fact_2509_binomial__code,axiom,
    ! [N: nat,K2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
       => ( aa(nat,nat,binomial(N),K2) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2)))
           => ( aa(nat,nat,binomial(N),K2) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2)))
           => ( aa(nat,nat,binomial(N),K2) = 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,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K2)) ) ) ) ) ) ).

% binomial_code
tff(fact_2510_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K2))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_2511_binomial__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa(nat,nat,binomial(N),K2) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2)) ) ).

% binomial_eq_0_iff
tff(fact_2512_binomial__Suc__Suc,axiom,
    ! [N: nat,K2: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K2))) ).

% binomial_Suc_Suc
tff(fact_2513_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_concat_bit(N,K2,L)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_2514_concat__bit__negative__iff,axiom,
    ! [N: nat,K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_concat_bit(N,K2,L)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ).

% concat_bit_negative_iff
tff(fact_2515_zero__less__binomial__iff,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ).

% zero_less_binomial_iff
tff(fact_2516_less__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),pred_numeral(K2)),N)) ) ).

% less_numeral_Suc
tff(fact_2517_less__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),pred_numeral(K2))) ) ).

% less_Suc_numeral
tff(fact_2518_le__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),pred_numeral(K2)),N)) ) ).

% le_numeral_Suc
tff(fact_2519_le__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),pred_numeral(K2))) ) ).

% le_Suc_numeral
tff(fact_2520_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one) = 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_2521_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one,aa(num,num,bit0,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one)) ) ).

% divmod_algorithm_code(3)
tff(fact_2522_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one)) ) ).

% divmod_algorithm_code(4)
tff(fact_2523_concat__bit__assoc,axiom,
    ! [N: nat,K2: int,M: nat,L: int,R3: int] : bit_concat_bit(N,K2,bit_concat_bit(M,L,R3)) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),bit_concat_bit(N,K2,L),R3) ).

% concat_bit_assoc
tff(fact_2524_binomial__eq__0,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
     => ( aa(nat,nat,binomial(N),K2) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2525_binomial__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => ( aa(nat,nat,binomial(N),K2) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) ) ).

% binomial_symmetric
tff(fact_2526_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R3)),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R3)),K2)),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R3)),M)) ).

% choose_mult_lemma
tff(fact_2527_binomial__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),R3)),aa(nat,nat,power_power(nat,N),R3))) ) ).

% binomial_le_pow
tff(fact_2528_zero__less__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K2))) ) ).

% zero_less_binomial
tff(fact_2529_Suc__times__binomial__add,axiom,
    ! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A3)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),aa(nat,nat,suc,A3))) = 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),A3),B2))),A3)) ).

% Suc_times_binomial_add
tff(fact_2530_choose__mult,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M)),aa(nat,nat,binomial(M),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K2))) ) ) ) ).

% choose_mult
tff(fact_2531_binomial__fact__lemma,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))),aa(nat,nat,binomial(N),K2)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_2532_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2)))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2533_choose__reduce__nat,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => ( aa(nat,nat,binomial(N),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_2534_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,binomial(N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2535_binomial__le__pow2,axiom,
    ! [N: nat,K2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))) ).

% binomial_le_pow2
tff(fact_2536_binomial__altdef__nat,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => ( aa(nat,nat,binomial(N),K2) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_2537_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_2538_binomial__addition__formula,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,suc,K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K2)) ) ) ).

% binomial_addition_formula
tff(fact_2539_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2))) ) ) ) ).

% fact_binomial
tff(fact_2540_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))) ) ) ) ).

% binomial_fact
tff(fact_2541_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,aa(num,num,bit0,N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_2542_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K2))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_2543_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A3,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2544_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2545_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,N)) = 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_dn(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_2546_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_2547_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))) ) ) ) ) ).

% take_bit_rec
tff(fact_2548_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),one_one(nat)))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_2549_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_left_iff
tff(fact_2550_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_right_iff
tff(fact_2551_dvd__1__left,axiom,
    ! [K2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K2)) ).

% dvd_1_left
tff(fact_2552_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_2553_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% mask_nat_positive_iff
tff(fact_2554_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),A3: 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),F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F3,A3),B2) ).

% case_prod_conv
tff(fact_2555_curry__case__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)) = F3 ).

% curry_case_prod
tff(fact_2556_case__prod__curry,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),F3)) = F3 ).

% case_prod_curry
tff(fact_2557_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2558_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2559_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) ) ) ) ) ).

% div_add
tff(fact_2560_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,B2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2561_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2)) ) ) ) ).

% odd_add
tff(fact_2562_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2)) ) ) ) ).

% even_add
tff(fact_2563_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)) ) ) ).

% even_plus_one_iff
tff(fact_2564_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ).

% even_diff
tff(fact_2565_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),W2)))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2566_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),W2)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2567_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A3),N)),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_2568_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ).

% even_diff_nat
tff(fact_2569_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_2570_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) ) ) ) ).

% even_succ_div_two
tff(fact_2571_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) ) ) ) ).

% even_succ_div_2
tff(fact_2572_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),W2)))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2573_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,A,power_power(A,A3),N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_2574_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,N)) = 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_do(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_2575_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_2576_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2)))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),W2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),W2)))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2577_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F3),X),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(D,fun(B,fun(C,A)),aTP_Lamp_dp(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F3),Y)),X) ).

% case_prod_app
tff(fact_2578_nested__case__prod__simp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F3),X),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(D,fun(B,fun(C,A)),aTP_Lamp_dp(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F3),Y)),X) ).

% nested_case_prod_simp
tff(fact_2579_prod_Ocase__distrib,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,H: fun(C,D),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(C,D,H,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)) = aa(product_prod(A,B),D,aa(fun(A,fun(B,D)),fun(product_prod(A,B),D),product_case_prod(A,B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_dq(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),H),F3)),Prod) ).

% prod.case_distrib
tff(fact_2580_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ).

% take_bit_add
tff(fact_2581_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_2582_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q5)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),Q5))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2583_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_2584_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ) ).

% dvd_add
tff(fact_2585_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_add_left_iff
tff(fact_2586_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% dvd_add_right_iff
tff(fact_2587_old_Oprod_Ocase,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,fun(B,C)),X1: A,X2: B] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2)) = aa(B,C,aa(A,fun(B,C),F3,X1),X2) ).

% old.prod.case
tff(fact_2588_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q5: product_prod(A,B),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C)),P3: product_prod(A,B)] :
      ( ! [X3: A,Y3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) = Q5 )
         => ( aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(B,C,aa(A,fun(B,C),G3,X3),Y3) ) )
     => ( ( P3 = Q5 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3),Q5) ) ) ) ).

% split_cong
tff(fact_2589_dvd__diff__nat,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

% dvd_diff_nat
tff(fact_2590_uminus__dvd__conv_I1_J,axiom,
    ! [D3: int,T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),T5))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,uminus_uminus(int),D3)),T5)) ) ).

% uminus_dvd_conv(1)
tff(fact_2591_uminus__dvd__conv_I2_J,axiom,
    ! [D3: int,T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),T5))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,uminus_uminus(int),T5))) ) ).

% uminus_dvd_conv(2)
tff(fact_2592_less__eq__mask,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),bit_se2239418461657761734s_mask(nat,N))) ).

% less_eq_mask
tff(fact_2593_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_2594_nat__take__bit__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(int,nat,nat2,K2)) ) ) ).

% nat_take_bit_eq
tff(fact_2595_take__bit__nat__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(int,nat,nat2,K2)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)) ) ) ).

% take_bit_nat_eq
tff(fact_2596_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q2: fun(A,bool),P2: fun(B,fun(C,A)),Z4: product_prod(B,C)] :
      ( pp(aa(A,bool,Q2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P2),Z4)))
     => ~ ! [X3: B,Y3: C] :
            ( ( Z4 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y3) )
           => ~ pp(aa(A,bool,Q2,aa(C,A,aa(B,fun(C,A),P2,X3),Y3))) ) ) ).

% case_prodE2
tff(fact_2597_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: 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_dr(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3)) = F3 ).

% case_prod_eta
tff(fact_2598_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),G3: fun(product_prod(A,B),C)] :
      ( ! [X3: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(product_prod(A,B),C,G3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) = G3 ) ) ).

% cond_case_prod_eta
tff(fact_2599_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ds(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ds(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_2600_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2) = bit_se2239418461657761734s_mask(int,N) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_2601_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ds(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ds(A,fun(A,bool),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_2602_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2) = bit_se2239418461657761734s_mask(int,N) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int)))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_2603_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K2)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2604_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),K2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_2605_take__bit__nonnegative,axiom,
    ! [N: nat,K2: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2))) ).

% take_bit_nonnegative
tff(fact_2606_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% take_bit_int_greater_self_iff
tff(fact_2607_not__take__bit__negative,axiom,
    ! [N: nat,K2: int] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),zero_zero(int))) ).

% not_take_bit_negative
tff(fact_2608_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X5))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% pinf(9)
tff(fact_2609_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X5))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% pinf(10)
tff(fact_2610_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X5),Z3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% minf(9)
tff(fact_2611_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S2: B] :
        ? [Z3: B] :
        ! [X5: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X5),Z3))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X5),S2))) ) ) ) ).

% minf(10)
tff(fact_2612_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = aa(A,A,if(fun(A,A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M),bit_se2584673776208193580ke_bit(A,N),bit_ri4674362597316999326ke_bit(A,M)),A3) ) ).

% signed_take_bit_take_bit
tff(fact_2613_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2614_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2615_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se2638667681897837118et_bit(A,M,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se2638667681897837118et_bit(A,M,A3)) = bit_se2638667681897837118et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) ) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_2616_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se5668285175392031749et_bit(A,M,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se5668285175392031749et_bit(A,M,A3)) = bit_se5668285175392031749et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) ) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_2617_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A3)) = bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) ) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_2618_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_2619_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).

% dvd_pos_nat
tff(fact_2620_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),N))) ) ) ).

% le_imp_power_dvd
tff(fact_2621_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat,B2: A,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,power_power(A,A3),N)),B2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,power_power(A,A3),M)),B2)) ) ) ) ).

% power_le_dvd
tff(fact_2622_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N: nat,M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,power_power(A,X),N)),aa(nat,A,power_power(A,Y),M))) ) ) ) ).

% dvd_power_le
tff(fact_2623_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% dvd_minus_self
tff(fact_2624_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),N))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2625_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N))
       => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M)) ) ) ).

% zdvd_not_zless
tff(fact_2626_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_2627_dvd__diffD1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N)) ) ) ) ).

% dvd_diffD1
tff(fact_2628_dvd__diffD,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),M)) ) ) ) ).

% dvd_diffD
tff(fact_2629_zdvd__mono,axiom,
    ! [K2: int,M: int,T5: int] :
      ( ( K2 != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),T5))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),T5))) ) ) ).

% zdvd_mono
tff(fact_2630_bezout__lemma__nat,axiom,
    ! [D3: nat,A3: nat,B2: nat,X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D3) )
            | ( 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),A3),Y)),D3) ) )
         => ? [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = 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),A3),B2)),Y3)),D3) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),D3) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_2631_bezout__add__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = 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),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_2632_zdvd__reduce,axiom,
    ! [K2: int,N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K2),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K2),M))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K2),N)) ) ).

% zdvd_reduce
tff(fact_2633_zdvd__period,axiom,
    ! [A3: int,D3: int,X: int,T5: int,C3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),D3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T5)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C3),D3))),T5))) ) ) ).

% zdvd_period
tff(fact_2634_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M))) ) ) ).

% fact_dvd
tff(fact_2635_uncurry__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : uncurry(A,B,C,F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) ).

% uncurry_def
tff(fact_2636_internal__case__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] : produc5280177257484947105e_prod(A,B,C) = product_case_prod(A,B,C) ).

% internal_case_prod_def
tff(fact_2637_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,N))) ).

% mask_nonnegative_int
tff(fact_2638_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2239418461657761734s_mask(int,N)),zero_zero(int))) ).

% not_mask_negative_int
tff(fact_2639_nat__dvd__iff,axiom,
    ! [Z4: int,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,Z4)),M))
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z4),aa(nat,int,semiring_1_of_nat(int),M))) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_2640_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2641_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P2: fun(A,bool),L: A] :
          ( ? [X4: A] : pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4)))
        <=> ? [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A))))
              & pp(aa(A,bool,P2,X4)) ) ) ) ).

% unity_coeff_ex
tff(fact_2642_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,T5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D5))
         => ! [X5: A,K4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T5)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),T5))) ) ) ) ).

% inf_period(3)
tff(fact_2643_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,T5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D5))
         => ! [X5: A,K4: A] :
              ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T5)))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),T5))) ) ) ) ).

% inf_period(4)
tff(fact_2644_dvd__imp__le,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ) ).

% dvd_imp_le
tff(fact_2645_dvd__mult__cancel,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% dvd_mult_cancel
tff(fact_2646_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2647_bezout__add__strong__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D2: nat,X3: nat,Y3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = 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_2648_zdvd__imp__le,axiom,
    ! [Z4: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z4),N))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),N)) ) ) ).

% zdvd_imp_le
tff(fact_2649_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,N))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)))) ) ).

% fact_fact_dvd_fact
tff(fact_2650_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2651_dvd__imp__le__int,axiom,
    ! [I2: int,D3: int] :
      ( ( I2 != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),I2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),D3)),aa(int,int,abs_abs(int),I2))) ) ) ).

% dvd_imp_le_int
tff(fact_2652_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( ( modulo_modulo(nat,M,Q5) = modulo_modulo(nat,N,Q5) )
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Q5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2653_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_2654_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),bit_se2239418461657761734s_mask(nat,N))) ) ).

% less_mask
tff(fact_2655_even__nat__iff,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(int,nat,nat2,K2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2)) ) ) ).

% even_nat_iff
tff(fact_2656_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_dt(nat,fun(nat,bool)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_2657_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% odd_even_add
tff(fact_2658_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),N)))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A)))
              | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_2659_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            | ( X = one_one(A) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(nat,A,power_power(A,X),N))) ) ) ).

% dvd_power
tff(fact_2660_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2661_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2662_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))),semiring_char_0_fact(A,N))) ) ) ).

% choose_dvd
tff(fact_2663_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one)))))) ) ).

% even_even_mod_4_iff
tff(fact_2664_dvd__minus__add,axiom,
    ! [Q5: nat,N: nat,R3: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q5),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q5),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q5)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)),Q5)))) ) ) ) ).

% dvd_minus_add
tff(fact_2665_mod__nat__eqI,axiom,
    ! [R3: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R3),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),R3)))
         => ( modulo_modulo(nat,M,N) = R3 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2666_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K2,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))
        | ( ( L = zero_zero(int) )
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) )
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L)) ) ) ).

% mod_int_pos_iff
tff(fact_2667_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,power_power(nat,I2),M)),aa(nat,nat,power_power(nat,I2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_2668_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_2669_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_2670_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2671_take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ).

% take_bit_int_less_exp
tff(fact_2672_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ).

% power_mono_odd
tff(fact_2673_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% odd_pos
tff(fact_2674_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),L)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L))) ) ).

% even_diff_iff
tff(fact_2675_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(int,int,abs_abs(int),L))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L))) ) ).

% even_add_abs_iff
tff(fact_2676_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K2)),L)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L))) ) ).

% even_abs_add_iff
tff(fact_2677_dvd__power__iff__le,axiom,
    ! [K2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,power_power(nat,K2),M)),aa(nat,nat,power_power(nat,K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_2678_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_2679_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),bit_se2239418461657761734s_mask(nat,N)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_2680_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2681_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),K2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),K2)) ) ).

% take_bit_int_less_self_iff
tff(fact_2682_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ~ ! [B4: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B4)),one_one(A)) ) ) ).

% oddE
tff(fact_2683_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2684_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% zero_le_odd_power
tff(fact_2685_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),N))) ) ) ).

% zero_le_even_power
tff(fact_2686_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ).

% power_mono_even
tff(fact_2687_take__bit__int__eq__self,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_2688_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2) = K2 )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2689_take__bit__incr__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2) != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)) ) ) ).

% take_bit_incr_eq
tff(fact_2690_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),N)))
        <=> ( ( N = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2691_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_2692_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))) ) ).

% take_bit_Suc
tff(fact_2693_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),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_du(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_2694_take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_2695_take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2))) ) ).

% take_bit_int_greater_eq
tff(fact_2696_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)),one_one(A))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N))))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% even_mask_div_iff'
tff(fact_2697_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K2: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_2698_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A3),N)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2699_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_2700_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: 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,K2))) = 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),K2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_2701_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_dv(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_2702_take__bit__minus__small__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2703_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)),one_one(A))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N))))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N) = zero_zero(A) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_2704_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one)) )
     => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_2705_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K2))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_2706_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            | ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N) = zero_zero(A) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2707_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_dw(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_2708_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),aTP_Lamp_dx(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_2709_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = 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_dy(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_2710_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))))) ) ).

% flip_bit_0
tff(fact_2711_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_2712_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(aa(num,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_2713_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( ( K2 = zero_zero(int) )
          | ( L = zero_zero(int) ) )
       => ( bit_se5824344872417868541ns_and(int,K2,L) = zero_zero(int) ) )
      & ( ~ ( ( K2 = zero_zero(int) )
            | ( L = zero_zero(int) ) )
       => ( ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( bit_se5824344872417868541ns_and(int,K2,L) = L ) )
          & ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( bit_se5824344872417868541ns_and(int,K2,L) = K2 ) )
              & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( bit_se5824344872417868541ns_and(int,K2,L) = 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,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_2714_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),C3: fun(A,fun(B,bool))] :
      ( ! [A5: A,B4: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
         => pp(aa(B,bool,aa(A,fun(B,bool),C3,A5),B4)) )
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C3),P3)) ) ).

% case_prodI2
tff(fact_2715_case__prodI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),F3,A3),B2))
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% case_prodI
tff(fact_2716_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),Z4: C,C3: fun(A,fun(B,set(C)))] :
      ( ! [A5: A,B4: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
         => pp(aa(set(C),bool,member(C,Z4),aa(B,set(C),aa(A,fun(B,set(C)),C3,A5),B4))) )
     => pp(aa(set(C),bool,member(C,Z4),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)),C3),P3))) ) ).

% mem_case_prodI2
tff(fact_2717_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z4: A,C3: fun(B,fun(C,set(A))),A3: B,B2: C] :
      ( pp(aa(set(A),bool,member(A,Z4),aa(C,set(A),aa(B,fun(C,set(A)),C3,A3),B2)))
     => pp(aa(set(A),bool,member(A,Z4),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)),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)))) ) ).

% mem_case_prodI
tff(fact_2718_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: product_prod(A,B),C3: fun(A,fun(B,fun(C,bool))),X: C] :
      ( ! [A5: A,B4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) = P3 )
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C3,A5),B4),X)) )
     => pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C3),P3),X)) ) ).

% case_prodI2'
tff(fact_2719_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: bool,Q2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),aa(bool,A,zero_neq_one_of_bool(A),Q2)))
        <=> ( pp(P2)
           => pp(Q2) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_2720_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: bool,Q2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),aa(bool,A,zero_neq_one_of_bool(A),Q2)))
        <=> ( ~ pp(P2)
            & pp(Q2) ) ) ) ).

% of_bool_less_iff
tff(fact_2721_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P2: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P2)) = aa(bool,A,zero_neq_one_of_bool(A),P2) ) ).

% of_nat_of_bool
tff(fact_2722_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> pp(P2) ) ) ).

% zero_less_of_bool_iff
tff(fact_2723_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),one_one(A)))
        <=> ~ pp(P2) ) ) ).

% of_bool_less_one_iff
tff(fact_2724_and__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344872417868541ns_and(int,K2,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_2725_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344872417868541ns_and(int,K2,L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% and_negative_int_iff
tff(fact_2726_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_dz(num,option(num)),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_2727_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_2728_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_2729_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_dz(num,option(num)),bit_take_bit_num(pred_numeral(R3),M)) ).

% take_bit_num_simps(6)
tff(fact_2730_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_2731_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_2732_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one,bit1,bit_take_bit_num(pred_numeral(R3),M))) ).

% take_bit_num_simps(7)
tff(fact_2733_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_2734_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),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),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_2735_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) ) ).

% take_bit_of_exp
tff(fact_2736_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) ) ).

% take_bit_of_2
tff(fact_2737_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_2738_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),R2: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P2,R2) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_eb(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),P2),R2)))) ).

% same_fst_def
tff(fact_2739_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_2740_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_ed(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_2741_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z4: A,C3: fun(B,fun(C,set(A))),P3: product_prod(B,C)] :
      ( pp(aa(set(A),bool,member(A,Z4),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)),C3),P3)))
     => ~ ! [X3: B,Y3: C] :
            ( ( P3 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y3) )
           => ~ pp(aa(set(A),bool,member(A,Z4),aa(C,set(A),aa(B,fun(C,set(A)),C3,X3),Y3))) ) ) ).

% mem_case_prodE
tff(fact_2742_case__prodE,axiom,
    ! [A: $tType,B: $tType,C3: fun(A,fun(B,bool)),P3: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C3),P3))
     => ~ ! [X3: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),C3,X3),Y3)) ) ) ).

% case_prodE
tff(fact_2743_case__prodD,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),F3,A3),B2)) ) ).

% case_prodD
tff(fact_2744_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(A,fun(B,fun(C,bool))),P3: product_prod(A,B),Z4: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C3),P3),Z4))
     => ~ ! [X3: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C3,X3),Y3),Z4)) ) ) ).

% case_prodE'
tff(fact_2745_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(A,fun(B,fun(C,bool))),A3: A,B2: B,C3: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C3))
     => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R2,A3),B2),C3)) ) ).

% case_prodD'
tff(fact_2746_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P2))) ) ).

% zero_less_eq_of_bool
tff(fact_2747_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),one_one(A))) ) ).

% of_bool_less_eq_one
tff(fact_2748_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344872417868541ns_and(int,X,Y))) ) ).

% AND_lower
tff(fact_2749_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,X,Y)),X)) ) ).

% AND_upper1
tff(fact_2750_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,X,Y)),Y)) ) ).

% AND_upper2
tff(fact_2751_AND__upper1_H,axiom,
    ! [Y: int,Z4: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z4))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,Y,Ya)),Z4)) ) ) ).

% AND_upper1'
tff(fact_2752_AND__upper2_H,axiom,
    ! [Y: int,Z4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z4))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,X,Y)),Z4)) ) ) ).

% AND_upper2'
tff(fact_2753_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A6),B5))
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),B5)))) ) ).

% Collect_case_prod_mono
tff(fact_2754_execute__bind__case,axiom,
    ! [A: $tType,B: $tType,F3: heap_Time_Heap(B),G3: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_bind(B,A,F3,G3)),H) = case_option(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),none(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),product_case_prod(B,product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ef(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),G3)),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F3),H)) ).

% execute_bind_case
tff(fact_2755_map__to__set__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_to_set(A,B,M) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_eg(fun(A,option(B)),fun(A,fun(B,bool)),M))) ).

% map_to_set_def
tff(fact_2756_and__int__rec,axiom,
    ! [K2: int,L: int] : bit_se5824344872417868541ns_and(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ).

% and_int_rec
tff(fact_2757_AND__upper2_H_H,axiom,
    ! [Y: int,Z4: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z4))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344872417868541ns_and(int,X,Y)),Z4)) ) ) ).

% AND_upper2''
tff(fact_2758_AND__upper1_H_H,axiom,
    ! [Y: int,Z4: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z4))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344872417868541ns_and(int,Y,Ya)),Z4)) ) ) ).

% AND_upper1''
tff(fact_2759_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se5824344872417868541ns_and(int,K2,L)),K2)) ) ).

% and_less_eq
tff(fact_2760_Heap__Time__Monad_Obind__def,axiom,
    ! [B: $tType,A: $tType,F3: heap_Time_Heap(A),G3: fun(A,heap_Time_Heap(B))] : heap_Time_bind(A,B,F3,G3) = heap_Time_Heap2(B,aa(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ej(heap_Time_Heap(A),fun(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),F3),G3)) ).

% Heap_Time_Monad.bind_def
tff(fact_2761_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P2: fun(product_prod(A,B),bool)] : rel_of(A,B,M,P2) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_ek(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),M),P2))) ).

% rel_of_def
tff(fact_2762_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P2: fun(A,bool),A3: A] :
          ( ! [A5: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A5),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = A5 )
             => pp(aa(A,bool,P2,A5)) )
         => ( ! [A5: A,B4: bool] :
                ( pp(aa(A,bool,P2,A5))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A5))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) = A5 )
                 => pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A5)))) ) )
           => pp(aa(A,bool,P2,A3)) ) ) ) ).

% bits_induct
tff(fact_2763_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : modulo_modulo(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)) ) ).

% exp_mod_exp
tff(fact_2764_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ).

% exp_div_exp_eq
tff(fact_2765_divide__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K2) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K2) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) )
          & ( ( aa(int,int,sgn_sgn(int),K2) != aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_2766_divide__int__def,axiom,
    ! [L: int,K2: int] :
      ( ( ( L = zero_zero(int) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L) = zero_zero(int) ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K2))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K2) != aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),L) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K2))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_2767_mlex__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F3,R2) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_el(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F3),R2))) ).

% mlex_eq
tff(fact_2768_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(4)
tff(fact_2769_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : bit_se5824344872417868541ns_and(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(8)
tff(fact_2770_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      ( ( ( pp(aa(set(int),bool,member(int,K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
          & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( bit_se5824344872417868541ns_and(int,K2,L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L))))) ) )
      & ( ~ ( pp(aa(set(int),bool,member(int,K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( bit_se5824344872417868541ns_and(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) ) ) ).

% and_int.simps
tff(fact_2771_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( bit_se5824344872417868541ns_and(int,X,Xa2) = Y )
     => ( ( ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),Xa2))))) ) )
        & ( ~ ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) ) ) ) ).

% and_int.elims
tff(fact_2772_Divides_Oadjust__div__eq,axiom,
    ! [Q5: int,R3: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q5),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R3),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_2773_insert__absorb2,axiom,
    ! [A: $tType,X: A,A6: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6) ).

% insert_absorb2
tff(fact_2774_insert__iff,axiom,
    ! [A: $tType,A3: A,B2: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A6)))
    <=> ( ( A3 = B2 )
        | pp(aa(set(A),bool,member(A,A3),A6)) ) ) ).

% insert_iff
tff(fact_2775_insertCI,axiom,
    ! [A: $tType,A3: A,B5: set(A),B2: A] :
      ( ( ~ pp(aa(set(A),bool,member(A,A3),B5))
       => ( A3 = B2 ) )
     => pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5))) ) ).

% insertCI
tff(fact_2776_split__part,axiom,
    ! [B: $tType,A: $tType,P2: bool,Q2: fun(A,fun(B,bool)),X5: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_em(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P2),Q2)),X5))
    <=> ( pp(P2)
        & pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q2),X5)) ) ) ).

% split_part
tff(fact_2777_singletonI,axiom,
    ! [A: $tType,A3: A] : pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) ).

% singletonI
tff(fact_2778_insert__subset,axiom,
    ! [A: $tType,X: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),B5))
    <=> ( pp(aa(set(A),bool,member(A,X),B5))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ) ).

% insert_subset
tff(fact_2779_Int__insert__right__if1,axiom,
    ! [A: $tType,A3: A,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ) ) ).

% Int_insert_right_if1
tff(fact_2780_Int__insert__right__if0,axiom,
    ! [A: $tType,A3: A,A6: set(A),B5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),A6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) ) ) ).

% Int_insert_right_if0
tff(fact_2781_insert__inter__insert,axiom,
    ! [A: $tType,A3: A,A6: set(A),B5: 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(A,fun(set(A),set(A)),insert(A),A3),A6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ).

% insert_inter_insert
tff(fact_2782_Int__insert__left__if1,axiom,
    ! [A: $tType,A3: A,C4: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),C4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ) ) ).

% Int_insert_left_if1
tff(fact_2783_Int__insert__left__if0,axiom,
    ! [A: $tType,A3: A,C4: set(A),B5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),C4))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4) ) ) ).

% Int_insert_left_if0
tff(fact_2784_Un__insert__left,axiom,
    ! [A: $tType,A3: A,B5: set(A),C4: 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(A,fun(set(A),set(A)),insert(A),A3),B5)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) ).

% Un_insert_left
tff(fact_2785_Un__insert__right,axiom,
    ! [A: $tType,A6: set(A),A3: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) ).

% Un_insert_right
tff(fact_2786_insert__Diff1,axiom,
    ! [A: $tType,X: A,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) ) ) ).

% insert_Diff1
tff(fact_2787_Diff__insert0,axiom,
    ! [A: $tType,X: A,A6: set(A),B5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) ) ) ).

% Diff_insert0
tff(fact_2788_singleton__conv,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ap(A,fun(A,bool),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ).

% singleton_conv
tff(fact_2789_singleton__conv2,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),fequal(A),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_2790_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A3: A,A6: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
    <=> ( ( A3 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2791_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A3: A,A6: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6) )
    <=> ( ( A3 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2792_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A6: set(A),B2: A,B5: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5)) )
    <=> ( ~ pp(aa(set(A),bool,member(A,B2),A6))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) ) ) ) ).

% disjoint_insert(2)
tff(fact_2793_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B5: set(A),A3: A,A6: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6)) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),B5))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A6) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_2794_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A3: A,A6: set(A),B5: 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),aa(A,fun(set(A),set(A)),insert(A),A3),A6)),B5) )
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),B5))
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) ) ) ) ).

% insert_disjoint(2)
tff(fact_2795_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A3: A,A6: set(A),B5: 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(A,fun(set(A),set(A)),insert(A),A3),A6)),B5) = bot_bot(set(A)) )
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),B5))
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_2796_insert__Diff__single,axiom,
    ! [A: $tType,A3: A,A6: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6) ).

% insert_Diff_single
tff(fact_2797_subset__Compl__singleton,axiom,
    ! [A: $tType,A6: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
    <=> ~ pp(aa(set(A),bool,member(A,B2),A6)) ) ).

% subset_Compl_singleton
tff(fact_2798_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_en(A,fun(B,bool))),Prod)) ).

% prod.disc_eq_case
tff(fact_2799_mk__disjoint__insert,axiom,
    ! [A: $tType,A3: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ? [B8: set(A)] :
          ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B8) )
          & ~ pp(aa(set(A),bool,member(A,A3),B8)) ) ) ).

% mk_disjoint_insert
tff(fact_2800_insert__commute,axiom,
    ! [A: $tType,X: A,Y: A,A6: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),A6)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) ).

% insert_commute
tff(fact_2801_insert__eq__iff,axiom,
    ! [A: $tType,A3: A,A6: set(A),B2: A,B5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),A6))
     => ( ~ pp(aa(set(A),bool,member(A,B2),B5))
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5) )
        <=> ( ( ( A3 = B2 )
             => ( A6 = B5 ) )
            & ( ( A3 != B2 )
             => ? [C6: set(A)] :
                  ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),C6) )
                  & ~ pp(aa(set(A),bool,member(A,B2),C6))
                  & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),C6) )
                  & ~ pp(aa(set(A),bool,member(A,A3),C6)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2802_insert__absorb,axiom,
    ! [A: $tType,A3: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6) = A6 ) ) ).

% insert_absorb
tff(fact_2803_insert__ident,axiom,
    ! [A: $tType,X: A,A6: set(A),B5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A6))
     => ( ~ pp(aa(set(A),bool,member(A,X),B5))
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B5) )
        <=> ( A6 = B5 ) ) ) ) ).

% insert_ident
tff(fact_2804_Set_Oset__insert,axiom,
    ! [A: $tType,X: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),A6))
     => ~ ! [B8: set(A)] :
            ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B8) )
           => pp(aa(set(A),bool,member(A,X),B8)) ) ) ).

% Set.set_insert
tff(fact_2805_insertI2,axiom,
    ! [A: $tType,A3: A,B5: set(A),B2: A] :
      ( pp(aa(set(A),bool,member(A,A3),B5))
     => pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5))) ) ).

% insertI2
tff(fact_2806_insertI1,axiom,
    ! [A: $tType,A3: A,B5: set(A)] : pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5))) ).

% insertI1
tff(fact_2807_insertE,axiom,
    ! [A: $tType,A3: A,B2: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A6)))
     => ( ( A3 != B2 )
       => pp(aa(set(A),bool,member(A,A3),A6)) ) ) ).

% insertE
tff(fact_2808_insert__compr,axiom,
    ! [A: $tType,A3: A,B5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_eo(A,fun(set(A),fun(A,bool)),A3),B5)) ).

% insert_compr
tff(fact_2809_insert__Collect,axiom,
    ! [A: $tType,A3: A,P2: fun(A,bool)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(fun(A,bool),set(A),collect(A),P2)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ep(A,fun(fun(A,bool),fun(A,bool)),A3),P2)) ).

% insert_Collect
tff(fact_2810_singleton__inject,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
     => ( A3 = B2 ) ) ).

% singleton_inject
tff(fact_2811_insert__not__empty,axiom,
    ! [A: $tType,A3: A,A6: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_2812_doubleton__eq__iff,axiom,
    ! [A: $tType,A3: A,B2: A,C3: A,D3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),D3),bot_bot(set(A)))) )
    <=> ( ( ( A3 = C3 )
          & ( B2 = D3 ) )
        | ( ( A3 = D3 )
          & ( B2 = C3 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2813_singleton__iff,axiom,
    ! [A: $tType,B2: A,A3: A] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))
    <=> ( B2 = A3 ) ) ).

% singleton_iff
tff(fact_2814_singletonD,axiom,
    ! [A: $tType,B2: A,A3: A] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))
     => ( B2 = A3 ) ) ).

% singletonD
tff(fact_2815_insert__mono,axiom,
    ! [A: $tType,C4: set(A),D5: set(A),A3: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),D5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),C4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),D5))) ) ).

% insert_mono
tff(fact_2816_subset__insert,axiom,
    ! [A: $tType,X: A,A6: set(A),B5: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B5)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ) ).

% subset_insert
tff(fact_2817_subset__insertI,axiom,
    ! [A: $tType,B5: set(A),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5))) ).

% subset_insertI
tff(fact_2818_subset__insertI2,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5))) ) ).

% subset_insertI2
tff(fact_2819_insert__subsetI,axiom,
    ! [A: $tType,X: A,A6: set(A),X6: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),X6)),A6)) ) ) ).

% insert_subsetI
tff(fact_2820_Int__insert__right,axiom,
    ! [A: $tType,A3: A,A6: set(A),B5: set(A)] :
      ( ( pp(aa(set(A),bool,member(A,A3),A6))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,A3),A6))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) ) ) ) ).

% Int_insert_right
tff(fact_2821_Int__insert__left,axiom,
    ! [A: $tType,A3: A,C4: set(A),B5: set(A)] :
      ( ( pp(aa(set(A),bool,member(A,A3),C4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,A3),C4))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4) ) ) ) ).

% Int_insert_left
tff(fact_2822_insert__Diff__if,axiom,
    ! [A: $tType,X: A,B5: set(A),A6: set(A)] :
      ( ( pp(aa(set(A),bool,member(A,X),B5))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),B5))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),B5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) ) ) ) ).

% insert_Diff_if
tff(fact_2823_Collect__conv__if,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( ( pp(aa(A,bool,P2,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_eq(fun(A,bool),fun(A,fun(A,bool)),P2),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P2,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_eq(fun(A,bool),fun(A,fun(A,bool)),P2),A3)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if
tff(fact_2824_Collect__conv__if2,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( ( pp(aa(A,bool,P2,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_er(fun(A,bool),fun(A,fun(A,bool)),P2),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) )
      & ( ~ pp(aa(A,bool,P2,A3))
       => ( aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_er(fun(A,bool),fun(A,fun(A,bool)),P2),A3)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if2
tff(fact_2825_insert__def,axiom,
    ! [A: $tType,A3: A,B5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ap(A,fun(A,bool),A3))),B5) ).

% insert_def
tff(fact_2826_subset__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),A3: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))
    <=> ( ( X6 = bot_bot(set(A)) )
        | ( X6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_2827_subset__singletonD,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))
     => ( ( A6 = bot_bot(set(A)) )
        | ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_2828_insert__is__Un,axiom,
    ! [A: $tType,A3: A,A6: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))),A6) ).

% insert_is_Un
tff(fact_2829_Un__singleton__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),X: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
    <=> ( ( ( A6 = bot_bot(set(A)) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
        | ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B5 = bot_bot(set(A)) ) )
        | ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_2830_singleton__Un__iff,axiom,
    ! [A: $tType,X: A,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) )
    <=> ( ( ( A6 = bot_bot(set(A)) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
        | ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B5 = bot_bot(set(A)) ) )
        | ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_2831_set__minus__singleton__eq,axiom,
    ! [A: $tType,X: A,X6: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),X6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X6 ) ) ).

% set_minus_singleton_eq
tff(fact_2832_insert__minus__eq,axiom,
    ! [A: $tType,X: A,Y: A,A6: set(A)] :
      ( ( X != Y )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_2833_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A6: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),A6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = A6 ) ) ).

% Diff_insert_absorb
tff(fact_2834_Diff__insert2,axiom,
    ! [A: $tType,A6: set(A),A3: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),B5) ).

% Diff_insert2
tff(fact_2835_insert__Diff,axiom,
    ! [A: $tType,A3: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = A6 ) ) ).

% insert_Diff
tff(fact_2836_Diff__insert,axiom,
    ! [A: $tType,A6: set(A),A3: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_2837_subset__Diff__insert,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),X: A,C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),C4))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C4)))
        & ~ pp(aa(set(A),bool,member(A,X),A6)) ) ) ).

% subset_Diff_insert
tff(fact_2838_subset__insert__iff,axiom,
    ! [A: $tType,A6: set(A),X: A,B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B5)))
    <=> ( ( pp(aa(set(A),bool,member(A,X),A6))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B5)) )
        & ( ~ pp(aa(set(A),bool,member(A,X),A6))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ) ) ).

% subset_insert_iff
tff(fact_2839_Diff__single__insert,axiom,
    ! [A: $tType,A6: set(A),X: A,B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B5))) ) ).

% Diff_single_insert
tff(fact_2840_remove__subset,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),S))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),S)) ) ).

% remove_subset
tff(fact_2841_Compl__insert,axiom,
    ! [A: $tType,X: A,A6: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_2842_psubset__insert__iff,axiom,
    ! [A: $tType,A6: set(A),X: A,B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B5)))
    <=> ( ( pp(aa(set(A),bool,member(A,X),B5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5)) )
        & ( ~ pp(aa(set(A),bool,member(A,X),B5))
         => ( ( pp(aa(set(A),bool,member(A,X),A6))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B5)) )
            & ( ~ pp(aa(set(A),bool,member(A,X),A6))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_2843_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = case_option(option(num),num,aa(num,option(num),some(num),one),aTP_Lamp_es(num,option(num)),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_2844_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_et(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_2845_and__nat__rec,axiom,
    ! [M: nat,N: nat] : bit_se5824344872417868541ns_and(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),M)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))) ).

% and_nat_rec
tff(fact_2846_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( bit_se5824344872417868541ns_and(nat,M,N) = zero_zero(nat) ) )
      & ( ~ ( ( M = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( bit_se5824344872417868541ns_and(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))) ) ) ) ).

% and_nat_unfold
tff(fact_2847_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( bit_se5824344872417868541ns_and(int,X,Xa2) = Y )
     => ( pp(aa(product_prod(int,int),bool,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),X),Xa2)))
       => ~ ( ( ( ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                  & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),Xa2))))) ) )
              & ( ~ ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,member(int,Xa2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,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),X),Xa2))) ) ) ) ).

% and_int.pelims
tff(fact_2848_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(product_prod(int,int),bool,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),L)))
     => ( ( ( pp(aa(set(int),bool,member(int,K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( bit_se5824344872417868541ns_and(int,K2,L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,member(int,K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,member(int,L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( bit_se5824344872417868541ns_and(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_2849_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : bit_se5824344872417868541ns_and(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_2850_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_2851_int__ge__less__than2__def,axiom,
    ! [D3: int] : int_ge_less_than2(D3) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_eu(int,fun(int,fun(int,bool)),D3))) ).

% int_ge_less_than2_def
tff(fact_2852_trancl__single,axiom,
    ! [A: $tType,A3: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),A3),B2)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),A3),B2)),bot_bot(set(product_prod(A,A)))) ).

% trancl_single
tff(fact_2853_trancl__sub__insert__trancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),X),R2)))) ).

% trancl_sub_insert_trancl
tff(fact_2854_one__plus__BitM,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one),bitM(N)) = aa(num,num,bit0,N) ).

% one_plus_BitM
tff(fact_2855_BitM__plus__one,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one) = aa(num,num,bit0,N) ).

% BitM_plus_one
tff(fact_2856_trancl__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),A3),B2)),R3)) = 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,R3)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_ev(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R3)))) ).

% trancl_insert2
tff(fact_2857_int__ge__less__than__def,axiom,
    ! [D3: int] : int_ge_less_than(D3) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_ew(int,fun(int,fun(int,bool)),D3))) ).

% int_ge_less_than_def
tff(fact_2858_sngr__assn__raw_Osimps,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),X: A,H: heap_ext(product_unit),As: set(nat)] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,R3,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
        <=> ( ( ref_get(A,H,R3) = X )
            & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,R3)),bot_bot(set(nat))) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,R3)),lim(product_unit,H))) ) ) ) ).

% sngr_assn_raw.simps
tff(fact_2859_sngr__assn__raw_Oelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
          ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb2))
          <=> pp(Y) )
         => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
               => ( pp(Y)
                <=> ~ ( ( ref_get(A,H2,X) = Xa2 )
                      & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H2))) ) ) ) ) ) ).

% sngr_assn_raw.elims(1)
tff(fact_2860_sngr__assn__raw_Oelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb2))
         => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
               => ~ ( ( ref_get(A,H2,X) = Xa2 )
                    & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H2))) ) ) ) ) ).

% sngr_assn_raw.elims(2)
tff(fact_2861_sngr__assn__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: ref(A),X: A] : sngr_assn(A,R3,X) = abs_assn(sngr_assn_raw(A,R3,X)) ) ).

% sngr_assn_def
tff(fact_2862_sngr__assn__raw_Oelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb2))
         => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
               => ( ( ref_get(A,H2,X) = Xa2 )
                  & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H2))) ) ) ) ) ).

% sngr_assn_raw.elims(3)
tff(fact_2863_sngr__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
          ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb2))
          <=> pp(Y) )
         => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
           => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                  ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
                 => ( ( pp(Y)
                    <=> ( ( ref_get(A,H2,X) = Xa2 )
                        & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H2))) ) )
                   => ~ pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))))) ) ) ) ) ) ).

% sngr_assn_raw.pelims(1)
tff(fact_2864_sngr__assn__raw_Opelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb2))
         => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
           => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                  ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
                 => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
                   => ~ ( ( ref_get(A,H2,X) = Xa2 )
                        & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H2))) ) ) ) ) ) ) ).

% sngr_assn_raw.pelims(2)
tff(fact_2865_sngr__assn__raw_Opelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: ref(A),Xa2: A,Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,sngr_assn_raw(A,X,Xa2),Xb2))
         => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
           => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                  ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
                 => ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
                   => ( ( ref_get(A,H2,X) = Xa2 )
                      & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_ref(A,X)),bot_bot(set(nat))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X)),lim(product_unit,H2))) ) ) ) ) ) ) ).

% sngr_assn_raw.pelims(3)
tff(fact_2866_log_Osimps,axiom,
    ! [B2: code_natural,I2: code_natural] :
      ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
          | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),B2)) )
       => ( log(B2,I2) = one_one(code_natural) ) )
      & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
            | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),B2)) )
       => ( log(B2,I2) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(B2,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),I2),B2))) ) ) ) ).

% log.simps
tff(fact_2867_log_Oelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
            | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
         => ( Y = one_one(code_natural) ) )
        & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
              | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
         => ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa2),X))) ) ) ) ) ).

% log.elims
tff(fact_2868_log_Opelims,axiom,
    ! [X: code_natural,Xa2: code_natural,Y: code_natural] :
      ( ( log(X,Xa2) = Y )
     => ( pp(aa(product_prod(code_natural,code_natural),bool,accp(product_prod(code_natural,code_natural),log_rel),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X),Xa2)))
       => ~ ( ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
                  | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
               => ( Y = one_one(code_natural) ) )
              & ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
                    | pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa2),X)) )
               => ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa2),X))) ) ) )
           => ~ pp(aa(product_prod(code_natural,code_natural),bool,accp(product_prod(code_natural,code_natural),log_rel),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X),Xa2))) ) ) ) ).

% log.pelims
tff(fact_2869_minus__shift__def,axiom,
    ! [K2: code_natural,L: code_natural,R3: code_natural] :
      ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K2),L))
       => ( minus_shift(R3,K2,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),R3),K2)),L) ) )
      & ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K2),L))
       => ( minus_shift(R3,K2,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K2),L) ) ) ) ).

% minus_shift_def
tff(fact_2870_next_Osimps,axiom,
    ! [V2: code_natural,W2: code_natural] : aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),V2),W2)) = aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one)))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W2),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one)))))))))))))))) ).

% next.simps
tff(fact_2871_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ex(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_2872_set__decode__plus__power__2,axiom,
    ! [N: nat,Z4: nat] :
      ( ~ pp(aa(set(nat),bool,member(nat,N),nat_set_decode(Z4)))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)),Z4)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N),nat_set_decode(Z4)) ) ) ).

% set_decode_plus_power_2
tff(fact_2873_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% subset_decode_imp_le
tff(fact_2874_take__bit__Suc__from__most,axiom,
    ! [N: nat,K2: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K2) = 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,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)) ).

% take_bit_Suc_from_most
tff(fact_2875_Random_Orange__def,axiom,
    ! [K2: code_natural] : range(K2) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),K2),aTP_Lamp_ez(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))))),one_one(code_natural)),aTP_Lamp_fa(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K2)) ).

% Random.range_def
tff(fact_2876_xor__Suc__0__eq,axiom,
    ! [N: nat] : bit_se5824344971392196577ns_xor(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))) ).

% xor_Suc_0_eq
tff(fact_2877_Suc__0__xor__eq,axiom,
    ! [N: nat] : bit_se5824344971392196577ns_xor(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))) ).

% Suc_0_xor_eq
tff(fact_2878_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list(bool)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(list(bool),int,aa(int,fun(list(bool),int),aa(fun(bool,int),fun(int,fun(list(bool),int)),groups4207007520872428315er_sum(bool,int),zero_neq_one_of_bool(int)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),Bs)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(list(bool),nat,size_size(list(bool)),Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_2879_scomp__apply,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(B,product_prod(C,D)),G3: fun(C,fun(D,A)),X: B] : aa(B,A,product_scomp(B,C,D,A,F3,G3),X) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),G3),aa(B,product_prod(C,D),F3,X)) ).

% scomp_apply
tff(fact_2880_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_2881_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),zero_zero(int)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_2882_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),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),aa(num,num,bit0,one))),bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_2883_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one))),bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_2884_scomp__scomp,axiom,
    ! [A: $tType,E: $tType,F2: $tType,B: $tType,D: $tType,C: $tType,F3: fun(A,product_prod(E,F2)),G3: fun(E,fun(F2,product_prod(C,D))),H: fun(C,fun(D,B))] : product_scomp(A,C,D,B,product_scomp(A,E,F2,product_prod(C,D),F3,G3),H) = product_scomp(A,E,F2,B,F3,aa(fun(C,fun(D,B)),fun(E,fun(F2,B)),aTP_Lamp_fb(fun(E,fun(F2,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F2,B))),G3),H)) ).

% scomp_scomp
tff(fact_2885_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A,N: nat] :
          ( ! [N4: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N4))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N4)) )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_2886_scomp__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,product_prod(B,C))] : product_scomp(A,B,C,product_prod(B,C),X,product_Pair(B,C)) = X ).

% scomp_Pair
tff(fact_2887_Pair__scomp,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: C,F3: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X),F3) = aa(C,fun(A,B),F3,X) ).

% Pair_scomp
tff(fact_2888_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ) ).

% bit_take_bit_iff
tff(fact_2889_scomp__def,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,F3: fun(A,product_prod(B,C)),G3: fun(B,fun(C,D)),X5: A] : aa(A,D,product_scomp(A,B,C,D,F3,G3),X5) = aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G3),aa(A,product_prod(B,C),F3,X5)) ).

% scomp_def
tff(fact_2890_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list(bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),Bs)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(bool),nat,size_size(list(bool)),Bs)))
            & pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_2891_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K2: code_natural,F3: fun(B,fun(A,product_prod(B,A))),X: B] :
      ( ( ( K2 = zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K2,F3),X) = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X) ) )
      & ( ( K2 != zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K2,F3),X) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F3,X),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K2),one_one(code_natural)),F3)) ) ) ) ).

% iterate.simps
tff(fact_2892_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb2: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa2),Xb2) = Y )
     => ( ( ( X = zero_zero(code_natural) )
         => ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb2) ) )
        & ( ( X != zero_zero(code_natural) )
         => ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb2),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa2)) ) ) ) ) ).

% iterate.elims
tff(fact_2893_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K2))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_2894_bit__concat__bit__iff,axiom,
    ! [M: nat,K2: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_concat_bit(M,K2,L)),N))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)) )
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_2895_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N4: nat] :
          ( ! [M4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),M4))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),M4))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N4)) ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N4),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N4)) ) ) ) ).

% int_bit_bound
tff(fact_2896_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_2897_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,N: nat] :
          ( ! [J3: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J3)))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2))),N))
          <=> ( ( ( N = zero_zero(nat) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_2898_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( bit_se5824344971392196577ns_xor(nat,M,N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( bit_se5824344971392196577ns_xor(nat,M,N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_se5824344971392196577ns_xor(nat,M,N) = 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),aa(num,num,bit0,one)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),bit_se5824344971392196577ns_xor(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_2899_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : bit_se5824344971392196577ns_xor(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),M))),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),bit_se5824344971392196577ns_xor(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))) ).

% xor_nat_rec
tff(fact_2900_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5824344971392196577ns_xor(A,A3,one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)))) ) ).

% xor_one_eq
tff(fact_2901_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5824344971392196577ns_xor(A,one_one(A),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)))) ) ).

% one_xor_eq
tff(fact_2902_set__bit__eq,axiom,
    ! [N: nat,K2: int] : bit_se5668285175392031749et_bit(int,N,K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ).

% set_bit_eq
tff(fact_2903_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,X: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_2904_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb2: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa2),Xb2) = Y )
     => ( pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb2))))
       => ~ ( ( ( ( X = zero_zero(code_natural) )
               => ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb2) ) )
              & ( ( X != zero_zero(code_natural) )
               => ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb2),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa2)) ) ) )
           => ~ pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb2)))) ) ) ) ).

% iterate.pelims
tff(fact_2905_the__elem__eq,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ).

% the_elem_eq
tff(fact_2906_inc__shift__def,axiom,
    ! [V2: code_natural,K2: code_natural] :
      ( ( ( V2 = K2 )
       => ( inc_shift(V2,K2) = one_one(code_natural) ) )
      & ( ( V2 != K2 )
       => ( inc_shift(V2,K2) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),K2),one_one(code_natural)) ) ) ) ).

% inc_shift_def
tff(fact_2907_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_fc(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_2908_xor__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344971392196577ns_xor(int,K2,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_2909_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344971392196577ns_xor(int,K2,L)),zero_zero(int)))
    <=> ~ ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% xor_negative_int_iff
tff(fact_2910_less__eq__integer__code_I1_J,axiom,
    pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_eq_integer_code(1)
tff(fact_2911_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_2912_plus__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K2),zero_zero(code_integer)) = K2 ).

% plus_integer_code(1)
tff(fact_2913_sgn__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( ( K2 = zero_zero(code_integer) )
       => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K2) = zero_zero(code_integer) ) )
      & ( ( K2 != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K2) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
           => ( aa(code_integer,code_integer,sgn_sgn(code_integer),K2) = one_one(code_integer) ) ) ) ) ) ).

% sgn_integer_code
tff(fact_2914_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se5824344971392196577ns_xor(int,X,Y))) ) ) ).

% XOR_lower
tff(fact_2915_bit__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K2)),N))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)) ) ) ).

% bit_nat_iff
tff(fact_2916_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se5824344971392196577ns_xor(int,X,Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ) ) ) ).

% XOR_upper
tff(fact_2917_xor__int__rec,axiom,
    ! [K2: int,L: int] : bit_se5824344971392196577ns_xor(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2))),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344971392196577ns_xor(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ).

% xor_int_rec
tff(fact_2918_integer__of__int__code,axiom,
    ! [K2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
       => ( aa(int,code_integer,code_integer_of_int,K2) = aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K2))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
       => ( ( ( K2 = zero_zero(int) )
           => ( aa(int,code_integer,code_integer_of_int,K2) = zero_zero(code_integer) ) )
          & ( ( K2 != zero_zero(int) )
           => ( aa(int,code_integer,code_integer_of_int,K2) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),zero_zero(int)),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),aa(num,num,bit0,one))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))))),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),aa(num,num,bit0,one))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_2919_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( bit_se5824344971392196577ns_xor(int,K2,L) = bit_ri4277139882892585799ns_not(int,L) ) )
      & ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( bit_se5824344971392196577ns_xor(int,K2,L) = bit_ri4277139882892585799ns_not(int,K2) ) )
          & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( K2 = zero_zero(int) )
               => ( bit_se5824344971392196577ns_xor(int,K2,L) = L ) )
              & ( ( K2 != zero_zero(int) )
               => ( ( ( L = zero_zero(int) )
                   => ( bit_se5824344971392196577ns_xor(int,K2,L) = K2 ) )
                  & ( ( L != zero_zero(int) )
                   => ( bit_se5824344971392196577ns_xor(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344971392196577ns_xor(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_2920_is__singleton__the__elem,axiom,
    ! [A: $tType,A6: set(A)] :
      ( is_singleton(A,A6)
    <=> ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the_elem(A,A6)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_2921_integer__of__num_I3_J,axiom,
    ! [N: num] : code_integer_of_num(aa(num,num,bit1,N)) = 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),plus_plus(code_integer),code_integer_of_num(N)),code_integer_of_num(N))),one_one(code_integer)) ).

% integer_of_num(3)
tff(fact_2922_take__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,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_fg(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M)) ).

% take_bit_num_code
tff(fact_2923_is__singletonI,axiom,
    ! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_2924_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,A),V2: num] : case_nat(A,A3,F3,aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,F3,pred_numeral(V2)) ).

% case_nat_numeral
tff(fact_2925_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_ri4277139882892585799ns_not(int,K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% not_nonnegative_int_iff
tff(fact_2926_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_ri4277139882892585799ns_not(int,K2)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% not_negative_int_iff
tff(fact_2927_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,A),V2: num,N: nat] : case_nat(A,A3,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N)) ).

% case_nat_add_eq_if
tff(fact_2928_abs__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K2) = aa(code_integer,code_integer,uminus_uminus(code_integer),K2) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K2) = K2 ) ) ) ).

% abs_integer_code
tff(fact_2929_less__integer__code_I1_J,axiom,
    ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_integer_code(1)
tff(fact_2930_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Xa2),X)) ) ).

% less_integer.abs_eq
tff(fact_2931_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(nat,A),Nat: nat] : aa(A,B,H,case_nat(A,F1,F22,Nat)) = case_nat(B,aa(A,B,H,F1),aa(fun(nat,A),fun(nat,B),aTP_Lamp_fh(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_2932_num_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(num,A),F32: fun(num,A),Num: num] : aa(A,B,H,case_num(A,F1,F22,F32,Num)) = case_num(B,aa(A,B,H,F1),aa(fun(num,A),fun(num,B),aTP_Lamp_fi(fun(A,B),fun(fun(num,A),fun(num,B)),H),F22),aa(fun(num,A),fun(num,B),aTP_Lamp_fi(fun(A,B),fun(fun(num,A),fun(num,B)),H),F32),Num) ).

% num.case_distrib
tff(fact_2933_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_2934_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_2935_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : bit_ri4277139882892585799ns_not(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_ri4277139882892585799ns_not(A,A3)),B2) ) ).

% not_add_distrib
tff(fact_2936_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : bit_ri4277139882892585799ns_not(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_ri4277139882892585799ns_not(A,A3)),B2) ) ).

% not_diff_distrib
tff(fact_2937_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X)) ).

% plus_integer.abs_eq
tff(fact_2938_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Xa2),X)) ) ).

% less_eq_integer.abs_eq
tff(fact_2939_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X2: num] : case_num(A,F1,F22,F32,aa(num,num,bit0,X2)) = aa(num,A,F22,X2) ).

% verit_eq_simplify(17)
tff(fact_2940_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : case_num(A,F1,F22,F32,one) = F1 ).

% verit_eq_simplify(16)
tff(fact_2941_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X32: num] : case_num(A,F1,F22,F32,aa(num,num,bit1,X32)) = aa(num,A,F32,X32) ).

% verit_eq_simplify(18)
tff(fact_2942_is__singletonI_H,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ( A6 != bot_bot(set(A)) )
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => ( pp(aa(set(A),bool,member(A,Y3),A6))
             => ( X3 = Y3 ) ) )
       => is_singleton(A,A6) ) ) ).

% is_singletonI'
tff(fact_2943_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_ri4277139882892585799ns_not(A,A3)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_2944_nth__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = case_nat(A,X,nth(A,Xs),N) ).

% nth_Cons
tff(fact_2945_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_ri4277139882892585799ns_not(A,bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_2946_is__singleton__def,axiom,
    ! [A: $tType,A6: set(A)] :
      ( is_singleton(A,A6)
    <=> ? [X4: A] : A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_2947_is__singletonE,axiom,
    ! [A: $tType,A6: set(A)] :
      ( is_singleton(A,A6)
     => ~ ! [X3: A] : A6 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_2948_integer__of__num_I2_J,axiom,
    ! [N: num] : code_integer_of_num(aa(num,num,bit0,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_num(N)),code_integer_of_num(N)) ).

% integer_of_num(2)
tff(fact_2949_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : bit_se5824344872417868541ns_and(int,bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_2950_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_2951_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,M)),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se5824344872417868541ns_and(int,aa(num,int,numeral_numeral(int),M),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_2952_not__int__rec,axiom,
    ! [K2: int] : bit_ri4277139882892585799ns_not(int,K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_ri4277139882892585799ns_not(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ).

% not_int_rec
tff(fact_2953_num__of__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),one_one(code_integer)))
       => ( code_num_of_integer(K2) = one ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),one_one(code_integer)))
       => ( code_num_of_integer(K2) = aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_fj(code_integer,fun(code_integer,num))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))) ) ) ) ).

% num_of_integer_code
tff(fact_2954_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
       => ( aa(code_integer,int,code_int_of_integer,K2) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K2))) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
       => ( ( ( K2 = zero_zero(code_integer) )
           => ( aa(code_integer,int,code_int_of_integer,K2) = zero_zero(int) ) )
          & ( ( K2 != zero_zero(code_integer) )
           => ( aa(code_integer,int,code_int_of_integer,K2) = aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_fk(code_integer,fun(code_integer,int))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_2955_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P2,case_nat(A,F1,F22,Nat)))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ pp(aa(A,bool,P2,F1)) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ pp(aa(A,bool,P2,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(2)
tff(fact_2956_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P2: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P2,case_nat(A,F1,F22,Nat)))
    <=> ( ( ( Nat = zero_zero(nat) )
         => pp(aa(A,bool,P2,F1)) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => pp(aa(A,bool,P2,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(1)
tff(fact_2957_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)),aa(nat,nat,suc,zero_zero(nat))) = groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_2958_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F3: fun(B,int),A6: set(B)] : aa(int,A,ring_1_of_int(A),groups7311177749621191930dd_sum(B,int,F3,A6)) = groups7311177749621191930dd_sum(B,A,aTP_Lamp_fl(fun(B,int),fun(B,A),F3),A6) ) ).

% of_int_sum
tff(fact_2959_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% plus_integer.rep_eq
tff(fact_2960_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F3: fun(B,nat),A6: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7311177749621191930dd_sum(B,nat,F3,A6)) = groups7311177749621191930dd_sum(B,A,aTP_Lamp_fm(fun(B,nat),fun(B,A),F3),A6) ) ).

% of_nat_sum
tff(fact_2961_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_fn(nat,nat),Nat) ).

% pred_def
tff(fact_2962_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A3: A,A6: set(B)] : modulo_modulo(A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_fo(fun(B,A),fun(A,fun(B,A)),F3),A3),A6),A3) = modulo_modulo(A,groups7311177749621191930dd_sum(B,A,F3,A6),A3) ) ).

% mod_sum_eq
tff(fact_2963_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_fp(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_2964_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_fq(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_2965_integer__less__iff,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,K2)),aa(code_integer,int,code_int_of_integer,L))) ) ).

% integer_less_iff
tff(fact_2966_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2))) ) ).

% less_integer.rep_eq
tff(fact_2967_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Y)))
    <=> pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Y)) ) ).

% int_of_integer_less_iff
tff(fact_2968_integer__less__eq__iff,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,K2)),aa(code_integer,int,code_int_of_integer,L))) ) ).

% integer_less_eq_iff
tff(fact_2969_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),X),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2))) ) ).

% less_eq_integer.rep_eq
tff(fact_2970_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I5: set(nat)] : groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fr(A,fun(nat,fun(nat,A)),X),M),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),groups7311177749621191930dd_sum(nat,A,power_power(A,X),I5)) ) ).

% sum_power_add
tff(fact_2971_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_2972_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_fn(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ).

% diff_Suc
tff(fact_2973_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W2))),N))
        <=> pp(case_nat(bool,fFalse,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),N)) ) ) ).

% bit_numeral_rec(1)
tff(fact_2974_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W2: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W2))),N))
        <=> pp(case_nat(bool,fTrue,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W2)),N)) ) ) ).

% bit_numeral_rec(2)
tff(fact_2975_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)),one_one(A)) = groups7311177749621191930dd_sum(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N))) ) ).

% mask_eq_sum_exp
tff(fact_2976_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A6: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),groups7311177749621191930dd_sum(A,B,aTP_Lamp_fs(fun(A,B),fun(A,B),F3),A6))) ) ).

% sum_abs_ge_zero
tff(fact_2977_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A6: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),groups7311177749621191930dd_sum(A,B,F3,A6))),groups7311177749621191930dd_sum(A,B,aTP_Lamp_fs(fun(A,B),fun(A,B),F3),A6))) ) ).

% sum_abs
tff(fact_2978_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,member(A,I3),I5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I3))) )
         => ( ( groups7311177749621191930dd_sum(A,B,X,I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( pp(aa(set(A),bool,member(A,I3),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I3)),B2))),Delta)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ft(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2979_abs__sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A6: set(A)] : aa(B,B,abs_abs(B),groups7311177749621191930dd_sum(A,B,aTP_Lamp_fs(fun(A,B),fun(A,B),F3),A6)) = groups7311177749621191930dd_sum(A,B,aTP_Lamp_fs(fun(A,B),fun(A,B),F3),A6) ) ).

% abs_sum_abs
tff(fact_2980_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B)] : groups7311177749621191930dd_sum(B,A,aTP_Lamp_fu(B,A),A6) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2981_int__sum,axiom,
    ! [B: $tType,F3: fun(B,nat),A6: set(B)] : aa(nat,int,semiring_1_of_nat(int),groups7311177749621191930dd_sum(B,nat,F3,A6)) = groups7311177749621191930dd_sum(B,int,aTP_Lamp_fv(fun(B,nat),fun(B,int),F3),A6) ).

% int_sum
tff(fact_2982_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,fun(C,A)),B5: set(C),A6: set(B)] : groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aTP_Lamp_fw(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B5),A6) = groups7311177749621191930dd_sum(C,A,aa(set(B),fun(C,A),aTP_Lamp_fy(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G3),A6),B5) ) ).

% sum.swap
tff(fact_2983_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),K5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7311177749621191930dd_sum(B,A,F3,K5)),groups7311177749621191930dd_sum(B,A,G3,K5))) ) ) ).

% sum_mono
tff(fact_2984_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),H: fun(B,A),A6: set(B)] : groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fz(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,A6)),groups7311177749621191930dd_sum(B,A,H,A6)) ) ).

% sum.distrib
tff(fact_2985_sum__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_0(B)
     => ! [F3: fun(A,B),A6: set(A),G3: fun(C,B),B5: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),groups7311177749621191930dd_sum(A,B,F3,A6)),groups7311177749621191930dd_sum(C,B,G3,B5)) = groups7311177749621191930dd_sum(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_gb(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F3),G3),B5),A6) ) ).

% sum_product
tff(fact_2986_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),A6: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),groups7311177749621191930dd_sum(B,A,F3,A6)),R3) = groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_gc(fun(B,A),fun(A,fun(B,A)),F3),R3),A6) ) ).

% sum_distrib_right
tff(fact_2987_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R3: A,F3: fun(B,A),A6: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R3),groups7311177749621191930dd_sum(B,A,F3,A6)) = groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gd(A,fun(fun(B,A),fun(B,A)),R3),F3),A6) ) ).

% sum_distrib_left
tff(fact_2988_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] : groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ge(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3),A6) = aa(A,A,aa(A,fun(A,A),minus_minus(A),groups7311177749621191930dd_sum(B,A,F3,A6)),groups7311177749621191930dd_sum(B,A,G3,A6)) ) ).

% sum_subtractf
tff(fact_2989_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),A6: set(B)] : groups7311177749621191930dd_sum(B,A,aTP_Lamp_gf(fun(B,A),fun(B,A),F3),A6) = aa(A,A,uminus_uminus(A),groups7311177749621191930dd_sum(B,A,F3,A6)) ) ).

% sum_negf
tff(fact_2990_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),A6: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7311177749621191930dd_sum(B,A,F3,A6)),R3) = groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_gg(fun(B,A),fun(A,fun(B,A)),F3),R3),A6) ) ).

% sum_divide_distrib
tff(fact_2991_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7311177749621191930dd_sum(B,A,F3,A6)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_2992_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),groups7311177749621191930dd_sum(B,A,F3,A6))) ) ) ).

% sum_nonneg
tff(fact_2993_sum__subtractf__nat,axiom,
    ! [A: $tType,A6: set(A),G3: fun(A,nat),F3: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G3,X3)),aa(A,nat,F3,X3))) )
     => ( groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_gh(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G3),F3),A6) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),groups7311177749621191930dd_sum(A,nat,F3,A6)),groups7311177749621191930dd_sum(A,nat,G3,A6)) ) ) ).

% sum_subtractf_nat
tff(fact_2994_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G3: fun(C,B),A6: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X3: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y3))
           => ( groups7311177749621191930dd_sum(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,H),G3),A6) = aa(B,A,H,groups7311177749621191930dd_sum(C,B,G3,A6)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_2995_sum__SucD,axiom,
    ! [A: $tType,F3: fun(A,nat),A6: set(A),N: nat] :
      ( ( groups7311177749621191930dd_sum(A,nat,F3,A6) = aa(nat,nat,suc,N) )
     => ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X3))) ) ) ).

% sum_SucD
tff(fact_2996_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),zero_zero(code_integer)))
       => ( code_nat_of_integer(K2) = zero_zero(nat) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),zero_zero(code_integer)))
       => ( code_nat_of_integer(K2) = aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_gi(code_integer,fun(code_integer,nat))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))) ) ) ) ).

% nat_of_integer_code
tff(fact_2997_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gj(nat,fun(nat,A),N),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N) ) ) ) ).

% choose_odd_sum
tff(fact_2998_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gk(nat,fun(nat,A),N),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N) ) ) ) ).

% choose_even_sum
tff(fact_2999_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gl(A,fun(nat,A),A3),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),aa(num,num,bit0,one)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_3000_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( ( ( X = one_one(A) )
               => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)) ) )
              & ( ( X != one_one(A) )
               => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_3001_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L4: A,H5: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L4,H5) )
        <=> ( ( ( L = L4 )
              & ( H = H5 ) )
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
              & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L4),H5)) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_3002_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or1337092689740270186AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% atLeastAtMost_iff
tff(fact_3003_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_atMost(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),K2)) ) ) ).

% atMost_iff
tff(fact_3004_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% atLeastatMost_empty_iff
tff(fact_3005_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_3006_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_3007_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3008_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),X)),aa(A,set(A),set_ord_atMost(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% atMost_subset_iff
tff(fact_3009_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H5: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H5)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H),H5)) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_3010_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ).

% sum.atMost_Suc
tff(fact_3011_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),zero_zero(code_integer)))
     => ( code_nat_of_integer(K2) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_3012_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_3013_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L4: A,H5: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L4,H5))) ) ).

% not_Iic_le_Icc
tff(fact_3014_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_gm(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_3015_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3016_ex__nat__less,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N))
          & pp(aa(nat,bool,P2,M3)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P2,X4)) ) ) ).

% ex_nat_less
tff(fact_3017_all__nat__less,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N))
         => pp(aa(nat,bool,P2,M3)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P2,X4)) ) ) ).

% all_nat_less
tff(fact_3018_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_3019_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3020_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(nat,fun(nat,A)),G3),K2),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_3021_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))) ) ) ) ).

% sum_power_shift
tff(fact_3022_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
              | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
                & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3))
                  | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3)) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3023_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( set_or1337092689740270186AtMost(nat,M,N) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3024_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3025_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_3026_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat,M: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,N,M)) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_3027_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gq(nat,fun(nat,nat),M),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_3028_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_3029_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_3030_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,aa(nat,nat,suc,N))),groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3031_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3032_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% sum.atMost_Suc_shift
tff(fact_3033_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),I2: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gr(fun(nat,A),fun(nat,A),F3),aa(nat,set(nat),set_ord_atMost(nat),I2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,aa(nat,nat,suc,I2))) ) ).

% sum_telescope
tff(fact_3034_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3035_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gs(fun(nat,A),fun(nat,A),F3),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,aa(nat,nat,suc,N))),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_3036_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_gt(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_3037_sum__choose__lower,axiom,
    ! [R3: nat,N: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gu(nat,fun(nat,nat),R3),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R3),N))),N) ).

% sum_choose_lower
tff(fact_3038_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gv(nat,fun(nat,nat),N),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),N),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_3039_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gv(nat,fun(nat,nat),N),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),N),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_3040_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),A3: nat,B2: nat] : groups7311177749621191930dd_sum(nat,A,F3,set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_gw(fun(nat,A),fun(nat,fun(A,A)),F3),A3,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3041_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A),P3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))),groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_3042_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_3043_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gx(A,fun(nat,A),A3),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_3044_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : 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),G3),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_gy(nat,fun(nat,fun(nat,bool)),N)))) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ha(fun(nat,fun(nat,A)),fun(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.triangle_reindex_eq
tff(fact_3045_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hb(nat,fun(nat,fun(nat,nat)),M),N),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_3046_vandermonde,axiom,
    ! [M: nat,N: nat,R3: nat] : groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hc(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R3),aa(nat,set(nat),set_ord_atMost(nat),R3)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R3) ).

% vandermonde
tff(fact_3047_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_3048_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hd(fun(nat,A),fun(nat,A),F3),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M)),aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hd(fun(nat,A),fun(nat,A),F3),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_3049_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_he(fun(nat,A),fun(nat,A),F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,N)),aa(nat,A,F3,M)) ) ) ) ).

% sum_telescope''
tff(fact_3050_choose__row__sum,axiom,
    ! [N: nat] : groups7311177749621191930dd_sum(nat,nat,binomial(N),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N) ).

% choose_row_sum
tff(fact_3051_binomial,axiom,
    ! [A3: nat,B2: nat,N: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),N) = groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hf(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N),aa(nat,set(nat),set_ord_atMost(nat),N)) ).

% binomial
tff(fact_3052_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,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),aa(num,num,bit0,one))),N)))) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hg(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.in_pairs_0
tff(fact_3053_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),X))
     => ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),Y))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),code_nat_of_integer(X)),code_nat_of_integer(Y)))
        <=> pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Y)) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_3054_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hh(A,fun(nat,A),A3),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_3055_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,N: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hi(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% binomial_ring
tff(fact_3056_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),groups7311177749621191930dd_sum(nat,A,power_power(A,X),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3057_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),N) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hj(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% pochhammer_binomial_sum
tff(fact_3058_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hg(fun(nat,A),fun(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.in_pairs
tff(fact_3059_choose__square__sum,axiom,
    ! [N: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hk(nat,fun(nat,nat),N),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_3060_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),P3))
           => ( groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H),aa(nat,set(nat),set_ord_atMost(nat),P3)) = groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_3061_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y: 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_hn(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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_ho(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_3062_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hp(nat,fun(nat,A),K2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_3063_gauss__sum__nat,axiom,
    ! [N: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fn(nat,nat),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) ).

% gauss_sum_nat
tff(fact_3064_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
          & ( ( X != one_one(A) )
           => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp0
tff(fact_3065_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y: 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_hn(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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_hq(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3066_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hr(nat,fun(nat,A),N),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3067_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hs(nat,fun(nat,A),M),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_3068_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ht(A,fun(A,fun(nat,A)),A3),D3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))) ) ).

% double_arith_series
tff(fact_3069_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_3070_binomial__r__part__sum,axiom,
    ! [M: 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),aa(num,num,bit0,one))),M)),one_one(nat))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_3071_choose__linear__sum,axiom,
    ! [N: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hu(nat,fun(nat,nat),N),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_3072_arith__series__nat,axiom,
    ! [A3: nat,D3: nat,N: nat] : groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hv(nat,fun(nat,fun(nat,nat)),A3),D3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D3)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) ).

% arith_series_nat
tff(fact_3073_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fn(nat,nat),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) ).

% Sum_Icc_nat
tff(fact_3074_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hw(nat,fun(nat,A),N),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3075_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3076_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D3: A,N: nat] : groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hx(A,fun(A,fun(nat,A)),A3),D3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) ) ).

% arith_series
tff(fact_3077_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) ) ).

% gauss_sum
tff(fact_3078_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N: nat] :
          ( ( ( X = one_one(A) )
           => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_offset
tff(fact_3079_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))) ) ).

% gauss_sum_from_Suc_0
tff(fact_3080_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R3: A,M: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gl(A,fun(nat,A),R3),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(nat,A,gbinomial(A,R3),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_3081_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : 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),aa(num,num,bit0,one))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_3082_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z4: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z4,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z4),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_hy(A,fun(nat,A),Z4),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),aa(num,num,bit0,one))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3083_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( ( ( K2 = zero_zero(code_integer) )
       => ( code_divmod_integer(K2,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K2 != zero_zero(code_integer) )
       => ( ( ( L = zero_zero(code_integer) )
           => ( code_divmod_integer(K2,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K2) ) )
          & ( ( L != zero_zero(code_integer) )
           => ( code_divmod_integer(K2,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K2)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K2,L),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_hz(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K2,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_3084_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N))
     => ( groups7311177749621191930dd_sum(int,int,aTP_Lamp_ia(int,int),set_or1337092689740270186AtMost(int,M,N)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_3085_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fn(nat,nat),set_or7035219750837199246ssThan(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) ).

% Sum_Ico_nat
tff(fact_3086_sum__power2,axiom,
    ! [K2: nat] : groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2)),one_one(nat)) ).

% sum_power2
tff(fact_3087_of__nat__id,axiom,
    ! [N: nat] : aa(nat,nat,semiring_1_of_nat(nat),N) = N ).

% of_nat_id
tff(fact_3088_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),X: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),aa(C,B,F3,Y)) ).

% apsnd_conv
tff(fact_3089_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_ib(B,A),A6) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3090_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,nat),A6: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,F3,A6)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_ic(fun(B,nat),fun(B,A),F3),A6) ) ).

% of_nat_prod
tff(fact_3091_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F3: fun(B,int),A6: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,F3,A6)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_id(fun(B,int),fun(B,A),F3),A6) ) ).

% of_int_prod
tff(fact_3092_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or7035219750837199246ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% atLeastLessThan_iff
tff(fact_3093_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_3094_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_3095_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_3096_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,J2: A,M: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I2,J2)),set_or7035219750837199246ssThan(A,M,N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J2),I2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),I2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J2),N)) ) ) ) ) ).

% ivl_subset
tff(fact_3097_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,N: A,M: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),N))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I2,M)),set_or7035219750837199246ssThan(A,I2,N)) = set_or7035219750837199246ssThan(A,N,M) ) ) ) ).

% ivl_diff
tff(fact_3098_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_3099_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_3100_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_3101_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,A),fun(nat,fun(nat,A)),G3),K2),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_3102_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_3103_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,P3: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))),groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,N,P3))) = groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3104_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_3105_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
            <=> ( ( A3 = C3 )
                & ( B2 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_3106_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
             => ( A3 = C3 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_3107_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
             => ( B2 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_3108_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A3: B,C3: B,B2: B,D3: B,G3: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C3),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) ) )
             => ( groups7121269368397514597t_prod(B,A,G3,set_or7035219750837199246ssThan(B,A3,B2)) = groups7121269368397514597t_prod(B,A,H,set_or7035219750837199246ssThan(B,C3,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3109_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,fun(C,A)),B5: set(C),A6: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_ig(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B5),A6) = groups7121269368397514597t_prod(C,A,aa(set(B),fun(C,A),aTP_Lamp_ii(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G3),A6),B5) ) ).

% prod.swap
tff(fact_3110_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),H: fun(B,A),A6: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ij(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G3,A6)),groups7121269368397514597t_prod(B,A,H,A6)) ) ).

% prod.distrib
tff(fact_3111_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_3112_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_3113_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ik(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3),A6) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F3,A6)),groups7121269368397514597t_prod(B,A,G3,A6)) ) ).

% prod_dividef
tff(fact_3114_prod__power__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [F3: fun(A,B),A6: set(A),N: nat] : aa(nat,B,power_power(B,groups7121269368397514597t_prod(A,B,F3,A6)),N) = groups7121269368397514597t_prod(A,B,aa(nat,fun(A,B),aTP_Lamp_il(fun(A,B),fun(nat,fun(A,B)),F3),N),A6) ) ).

% prod_power_distrib
tff(fact_3115_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_3116_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A3: A,A6: set(B)] : modulo_modulo(A,groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_fo(fun(B,A),fun(A,fun(B,A)),F3),A3),A6),A3) = modulo_modulo(A,groups7121269368397514597t_prod(B,A,F3,A6),A3) ) ).

% mod_prod_eq
tff(fact_3117_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,N)),groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% prod.last_plus
tff(fact_3118_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F3: fun(B,A),A6: set(B)] : aa(A,A,abs_abs(A),groups7121269368397514597t_prod(B,A,F3,A6)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_im(fun(B,A),fun(B,A),F3),A6) ) ).

% abs_prod
tff(fact_3119_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_gt(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_3120_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_in(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_3121_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_io(fun(nat,fun(nat,A)),fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_3122_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(C,B),G3: fun(D,C),X: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G3),X)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),comp(C,B,D,F3),G3)),X) ).

% apsnd_compose
tff(fact_3123_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),groups7121269368397514597t_prod(B,A,F3,A6))) ) ) ).

% prod_nonneg
tff(fact_3124_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7121269368397514597t_prod(B,A,F3,A6)),groups7121269368397514597t_prod(B,A,G3,A6))) ) ) ).

% prod_mono
tff(fact_3125_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% prod.head_if
tff(fact_3126_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),groups7121269368397514597t_prod(B,A,F3,A6))) ) ) ).

% prod_pos
tff(fact_3127_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),groups7121269368397514597t_prod(B,A,F3,A6))) ) ) ).

% prod_ge_1
tff(fact_3128_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_3129_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_3130_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3131_all__nat__less__eq,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
         => pp(aa(nat,bool,P2,M3)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P2,X4)) ) ) ).

% all_nat_less_eq
tff(fact_3132_ex__nat__less__eq,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
          & pp(aa(nat,bool,P2,M3)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P2,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_3133_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ir(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3134_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_is(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_3135_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_3136_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_3137_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3138_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C3: A,F3: fun(B,nat),A6: set(B)] : aa(nat,A,power_power(A,C3),groups7311177749621191930dd_sum(B,nat,F3,A6)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_it(A,fun(fun(B,nat),fun(B,A)),C3),F3),A6) ) ).

% power_sum
tff(fact_3139_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,A),fun(nat,fun(nat,A)),G3),K2),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3140_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),G3),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_3141_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,K2: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(nat,fun(nat,A)),G3),K2),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_3142_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),one_one(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7121269368397514597t_prod(B,A,F3,A6)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_3143_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2))) ) ) ) ).

% fact_split
tff(fact_3144_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A3: B,C3: B,B2: B,D3: B,G3: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C3),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) ) )
             => ( groups7311177749621191930dd_sum(B,A,G3,set_or7035219750837199246ssThan(B,A3,B2)) = groups7311177749621191930dd_sum(B,A,H,set_or7035219750837199246ssThan(B,C3,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_3145_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),K2) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iu(A,fun(nat,fun(nat,A)),A3),K2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_altdef_of_nat
tff(fact_3146_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iv(nat,fun(nat,fun(nat,A)),K2),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_3147_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3148_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,gbinomial(A,A3),K2)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_iw(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact
tff(fact_3149_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),semiring_char_0_fact(A,K2)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_iw(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact'
tff(fact_3150_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_ix(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2))),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_prod_rev
tff(fact_3151_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,P3: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,N,P3))) = groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_3152_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,P3: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),groups7311177749621191930dd_sum(nat,A,F3,set_or7035219750837199246ssThan(nat,M,P3))),groups7311177749621191930dd_sum(nat,A,F3,set_or7035219750837199246ssThan(nat,M,N))) = groups7311177749621191930dd_sum(nat,A,F3,set_or7035219750837199246ssThan(nat,N,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_3153_bset_I1_J,axiom,
    ! [D5: int,B5: set(int),P2: fun(int,bool),Q2: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,member(int,Xb),B5))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
         => ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,member(int,Xb),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
           => ( pp(aa(int,bool,Q2,X3))
             => pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P2,X5))
                & pp(aa(int,bool,Q2,X5)) )
             => ( pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)))
                & pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5))) ) ) ) ) ) ).

% bset(1)
tff(fact_3154_bset_I2_J,axiom,
    ! [D5: int,B5: set(int),P2: fun(int,bool),Q2: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,member(int,Xb),B5))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
         => ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,member(int,Xb),B5))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
           => ( pp(aa(int,bool,Q2,X3))
             => pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P2,X5))
                | pp(aa(int,bool,Q2,X5)) )
             => ( pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)))
                | pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5))) ) ) ) ) ) ).

% bset(2)
tff(fact_3155_aset_I1_J,axiom,
    ! [D5: int,A6: set(int),P2: fun(int,bool),Q2: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,member(int,Xb),A6))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
         => ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,member(int,Xb),A6))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
           => ( pp(aa(int,bool,Q2,X3))
             => pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A6))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P2,X5))
                & pp(aa(int,bool,Q2,X5)) )
             => ( pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)))
                & pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5))) ) ) ) ) ) ).

% aset(1)
tff(fact_3156_aset_I2_J,axiom,
    ! [D5: int,A6: set(int),P2: fun(int,bool),Q2: fun(int,bool)] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb: int] :
                  ( pp(aa(set(int),bool,member(int,Xb),A6))
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
         => ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb: int] :
                    ( pp(aa(set(int),bool,member(int,Xb),A6))
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
           => ( pp(aa(int,bool,Q2,X3))
             => pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))) ) )
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A6))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( pp(aa(int,bool,P2,X5))
                | pp(aa(int,bool,Q2,X5)) )
             => ( pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)))
                | pp(aa(int,bool,Q2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5))) ) ) ) ) ) ).

% aset(2)
tff(fact_3157_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( set_or7035219750837199246ssThan(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I2,J2)),set_or7035219750837199246ssThan(nat,J2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_3158_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ir(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3159_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3160_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,aa(nat,nat,suc,N))),groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3161_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3162_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3163_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_3164_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G3,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_3165_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_3166_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_3167_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_3168_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N)) = groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_3169_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,N)),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% sum.last_plus
tff(fact_3170_aset_I10_J,axiom,
    ! [D3: int,D5: int,A6: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),A6))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),T5)))
           => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)),T5))) ) ) ) ).

% aset(10)
tff(fact_3171_aset_I9_J,axiom,
    ! [D3: int,D5: int,A6: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),A6))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),T5)))
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)),T5))) ) ) ) ).

% aset(9)
tff(fact_3172_bset_I10_J,axiom,
    ! [D3: int,D5: int,B5: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),B5))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),T5)))
           => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)),T5))) ) ) ) ).

% bset(10)
tff(fact_3173_bset_I9_J,axiom,
    ! [D3: int,D5: int,B5: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),B5))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),T5)))
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)),T5))) ) ) ) ).

% bset(9)
tff(fact_3174_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3175_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3176_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G3,aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% prod.atMost_Suc_shift
tff(fact_3177_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_gt(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_3178_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N),set_or7035219750837199246ssThan(nat,M,N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_3179_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_fn(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_3180_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gs(fun(nat,A),fun(nat,A),F3),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,N)),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_3181_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_gt(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_3182_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat,M: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,N,M)) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_iy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_3183_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_iz(fun(nat,fun(nat,A)),fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_3184_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F3: fun(nat,A),A3: nat,B2: nat] : groups7121269368397514597t_prod(nat,A,F3,set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_jc(fun(nat,A),fun(nat,fun(A,A)),F3),A3,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3185_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_3186_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A),P3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N))),groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3187_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_3188_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% sum.head_if
tff(fact_3189_periodic__finite__ex,axiom,
    ! [D3: int,P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K: int] :
            ( pp(aa(int,bool,P2,X3))
          <=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) )
       => ( ? [X_12: int] : pp(aa(int,bool,P2,X_12))
        <=> ? [X4: int] :
              ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D3)))
              & pp(aa(int,bool,P2,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3190_bset_I3_J,axiom,
    ! [D5: int,T5: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T5),one_one(int))),B5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X5 = T5 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5) = T5 ) ) ) ) ) ).

% bset(3)
tff(fact_3191_bset_I4_J,axiom,
    ! [D5: int,T5: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,T5),B5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( ( X5 != T5 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5) != T5 ) ) ) ) ) ).

% bset(4)
tff(fact_3192_bset_I5_J,axiom,
    ! [D5: int,B5: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),B5))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X5),T5))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)),T5)) ) ) ) ).

% bset(5)
tff(fact_3193_bset_I7_J,axiom,
    ! [D5: int,T5: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,T5),B5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T5),X5))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T5),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5))) ) ) ) ) ).

% bset(7)
tff(fact_3194_aset_I3_J,axiom,
    ! [D5: int,T5: int,A6: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T5),one_one(int))),A6))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A6))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X5 = T5 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5) = T5 ) ) ) ) ) ).

% aset(3)
tff(fact_3195_aset_I4_J,axiom,
    ! [D5: int,T5: int,A6: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,T5),A6))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A6))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( ( X5 != T5 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5) != T5 ) ) ) ) ) ).

% aset(4)
tff(fact_3196_aset_I5_J,axiom,
    ! [D5: int,T5: int,A6: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,T5),A6))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A6))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X5),T5))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)),T5)) ) ) ) ) ).

% aset(5)
tff(fact_3197_aset_I7_J,axiom,
    ! [D5: int,A6: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),A6))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T5),X5))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T5),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5))) ) ) ) ).

% aset(7)
tff(fact_3198_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K2: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K2)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K2)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K2))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K2)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K2)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_3199_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_fn(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_3200_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat,M: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or7035219750837199246ssThan(nat,N,M)) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3201_simp__from__to,axiom,
    ! [J2: int,I2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I2))
       => ( set_or1337092689740270186AtMost(int,I2,J2) = bot_bot(set(int)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I2))
       => ( set_or1337092689740270186AtMost(int,I2,J2) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),I2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J2)) ) ) ) ).

% simp_from_to
tff(fact_3202_aset_I8_J,axiom,
    ! [D5: int,A6: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),A6))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T5),X5))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T5),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5))) ) ) ) ).

% aset(8)
tff(fact_3203_aset_I6_J,axiom,
    ! [D5: int,T5: int,A6: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T5),one_one(int))),A6))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),A6))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X5),T5))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),D5)),T5)) ) ) ) ) ).

% aset(6)
tff(fact_3204_bset_I8_J,axiom,
    ! [D5: int,T5: int,B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( pp(aa(set(int),bool,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T5),one_one(int))),B5))
       => ! [X5: int] :
            ( ! [Xa4: int] :
                ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
               => ! [Xb3: int] :
                    ( pp(aa(set(int),bool,member(int,Xb3),B5))
                   => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T5),X5))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T5),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5))) ) ) ) ) ).

% bset(8)
tff(fact_3205_bset_I6_J,axiom,
    ! [D5: int,B5: set(int),T5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ! [X5: int] :
          ( ! [Xa4: int] :
              ( pp(aa(set(int),bool,member(int,Xa4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
             => ! [Xb3: int] :
                  ( pp(aa(set(int),bool,member(int,Xb3),B5))
                 => ( X5 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X5),T5))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),D5)),T5)) ) ) ) ).

% bset(6)
tff(fact_3206_cpmi,axiom,
    ! [D5: int,P2: fun(int,bool),P4: fun(int,bool),B5: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z6))
           => ( pp(aa(int,bool,P2,X3))
            <=> pp(aa(int,bool,P4,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,member(int,Xb),B5))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
             => ( pp(aa(int,bool,P2,X3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5))) ) )
         => ( ! [X3: int,K: int] :
                ( pp(aa(int,bool,P4,X3))
              <=> pp(aa(int,bool,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D5)))) )
           => ( ? [X_12: int] : pp(aa(int,bool,P2,X_12))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & pp(aa(int,bool,P4,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,member(int,Xa3),B5))
                        & pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3207_cppi,axiom,
    ! [D5: int,P2: fun(int,bool),P4: fun(int,bool),A6: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( ? [Z6: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z6),X3))
           => ( pp(aa(int,bool,P2,X3))
            <=> pp(aa(int,bool,P4,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                 => ! [Xb: int] :
                      ( pp(aa(set(int),bool,member(int,Xb),A6))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
             => ( pp(aa(int,bool,P2,X3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))) ) )
         => ( ! [X3: int,K: int] :
                ( pp(aa(int,bool,P4,X3))
              <=> pp(aa(int,bool,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D5)))) )
           => ( ? [X_12: int] : pp(aa(int,bool,P2,X_12))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & pp(aa(int,bool,P4,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,member(int,X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,member(int,Xa3),A6))
                        & pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3208_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_is(A,fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_3209_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jd(A,fun(nat,fun(nat,A)),A3),N),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_3210_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_fn(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_3211_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_je(fun(nat,A),fun(nat,A),G3),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.in_pairs
tff(fact_3212_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G3,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),aa(num,num,bit0,one))),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_je(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.in_pairs_0
tff(fact_3213_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jd(A,fun(nat,fun(nat,A)),A3),N),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3214_divmod__integer__code,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( ( ( K2 = zero_zero(code_integer) )
       => ( code_divmod_integer(K2,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K2 != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K2))
               => ( code_divmod_integer(K2,L) = code_divmod_abs(K2,L) ) )
              & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K2))
               => ( code_divmod_integer(K2,L) = 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_jf(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K2,L)) ) ) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( ( L = zero_zero(code_integer) )
               => ( code_divmod_integer(K2,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K2) ) )
              & ( ( L != zero_zero(code_integer) )
               => ( code_divmod_integer(K2,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)),code_divmod_abs(K2,L),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_jg(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K2,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_3215_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_jh(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A3),Xs),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3216_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),P3))
           => ( groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ji(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H),aa(nat,set(nat),set_ord_atMost(nat),P3)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_jj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_3217_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_ix(A,fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K2))),semiring_char_0_fact(A,aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc
tff(fact_3218_bit__cut__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( ( K2 = zero_zero(code_integer) )
       => ( code_bit_cut_integer(K2) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),zero_zero(code_integer)),fFalse) ) )
      & ( ( K2 != zero_zero(code_integer) )
       => ( code_bit_cut_integer(K2) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,bool)),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool)),aTP_Lamp_jk(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K2)),code_divmod_abs(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_3219_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat))) ).

% natLess_def
tff(fact_3220_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : 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),G3),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jl(nat,fun(nat,fun(nat,bool)),N)))) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ha(fun(nat,fun(nat,A)),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.triangle_reindex
tff(fact_3221_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,fun(A,A)),V2: num,N: nat] : aa(nat,A,rec_nat(A,A3,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),N)) = aa(A,A,aa(nat,fun(A,A),F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N)),aa(nat,A,rec_nat(A,A3,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),N))) ).

% rec_nat_add_eq_if
tff(fact_3222_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K2))
       => ( nat_prod_decode_aux(K2,M) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),M)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K2))
       => ( nat_prod_decode_aux(K2,M) = nat_prod_decode_aux(aa(nat,nat,suc,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,K2))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_3223_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_lessThan(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),K2)) ) ) ).

% lessThan_iff
tff(fact_3224_old_Onat_Osimps_I6_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T))] : aa(nat,T,rec_nat(T,F1,F22),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_3225_old_Onat_Osimps_I7_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T)),Nat: nat] : aa(nat,T,rec_nat(T,F1,F22),aa(nat,nat,suc,Nat)) = aa(T,T,aa(nat,fun(T,T),F22,Nat),aa(nat,T,rec_nat(T,F1,F22),Nat)) ).

% old.nat.simps(7)
tff(fact_3226_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),X)),aa(A,set(A),set_ord_lessThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% lessThan_subset_iff
tff(fact_3227_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,fun(A,A)),V2: num] : aa(nat,A,rec_nat(A,A3,F3),aa(num,nat,numeral_numeral(nat),V2)) = aa(A,A,aa(nat,fun(A,A),F3,pred_numeral(V2)),aa(nat,A,rec_nat(A,A3,F3),pred_numeral(V2))) ).

% rec_nat_numeral
tff(fact_3228_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G3,N)) ) ).

% sum.lessThan_Suc
tff(fact_3229_int__prod,axiom,
    ! [B: $tType,F3: fun(B,nat),A6: set(B)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(B,nat,F3,A6)) = groups7121269368397514597t_prod(B,int,aTP_Lamp_fv(fun(B,nat),fun(B,int),F3),A6) ).

% int_prod
tff(fact_3230_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_jm(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_3231_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M)),aa(A,set(A),set_ord_lessThan(A),N)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N)) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3232_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_3233_rec__nat__0__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_3234_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N: nat] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),F22,N),aa(nat,A,F3,N)) ) ) ).

% rec_nat_Suc_imp
tff(fact_3235_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_3236_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(nat,A)),G3),N),aa(nat,set(nat),set_ord_lessThan(nat),N)) = groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nat_diff_reindex
tff(fact_3237_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jo(fun(nat,A),fun(nat,fun(nat,A)),G3),N),aa(nat,set(nat),set_ord_lessThan(nat),N)) = groups7121269368397514597t_prod(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nat_diff_reindex
tff(fact_3238_prod__int__eq,axiom,
    ! [I2: nat,J2: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I2,J2)) = groups7121269368397514597t_prod(int,int,aTP_Lamp_ia(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),J2))) ).

% prod_int_eq
tff(fact_3239_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_3240_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : 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),G3),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jl(nat,fun(nat,fun(nat,bool)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jq(fun(nat,fun(nat,A)),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.triangle_reindex
tff(fact_3241_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3242_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q2: fun(A,nat),P2: fun(A,nat),N: A] :
          ( ! [X3: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q2,X3)),aa(A,nat,P2,X3)))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),groups7311177749621191930dd_sum(A,nat,P2,aa(A,set(A),set_ord_lessThan(A),N))),groups7311177749621191930dd_sum(A,nat,Q2,aa(A,set(A),set_ord_lessThan(A),N))) = groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_jr(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q2),P2),aa(A,set(A),set_ord_lessThan(A),N)) ) ) ) ).

% sum_diff_distrib
tff(fact_3243_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3244_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gr(fun(nat,A),fun(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3245_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gs(fun(nat,A),fun(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M)),aa(nat,A,F3,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3246_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3247_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3248_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3249_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mm: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = groups7311177749621191930dd_sum(nat,A,F3,set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3250_prod__int__plus__eq,axiom,
    ! [I2: nat,J2: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2))) = groups7121269368397514597t_prod(int,int,aTP_Lamp_ia(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)))) ).

% prod_int_plus_eq
tff(fact_3251_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),K2: nat,N: nat] : groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_js(fun(nat,A),fun(nat,fun(nat,A)),G3),K2),aa(nat,set(nat),set_ord_lessThan(nat),N)) = groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2))) ) ).

% sum.nat_group
tff(fact_3252_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),K2: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jt(fun(nat,A),fun(nat,fun(nat,A)),G3),K2),aa(nat,set(nat),set_ord_lessThan(nat),N)) = groups7121269368397514597t_prod(nat,A,G3,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2))) ) ).

% prod.nat_group
tff(fact_3253_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ju(fun(nat,fun(nat,A)),fun(nat,A),A3),aa(nat,set(nat),set_ord_atMost(nat),N)) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nested_swap'
tff(fact_3254_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_jv(fun(nat,fun(nat,A)),fun(nat,A),A3),aa(nat,set(nat),set_ord_atMost(nat),N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nested_swap'
tff(fact_3255_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,K2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K2))
           => ( 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),K2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K2))
           => ( 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),K2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).

% Iio_Int_singleton
tff(fact_3256_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),aa(A,fun(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_3257_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_3258_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq
tff(fact_3259_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,X),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))),groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_1_eq
tff(fact_3260_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: nat] :
          ( ( X != one_one(A) )
         => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,X),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3261_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : groups7311177749621191930dd_sum(nat,A,G3,aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.atMost_shift
tff(fact_3262_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G3,aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.atMost_shift
tff(fact_3263_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
          & ( ( X != one_one(A) )
           => ( groups7311177749621191930dd_sum(nat,A,power_power(A,X),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,X),N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_strict
tff(fact_3264_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,X),N)),aa(nat,A,power_power(A,Y),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_jw(A,fun(nat,fun(A,fun(nat,A))),X),N),Y),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_sumr2
tff(fact_3265_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))),aa(nat,A,power_power(A,Y),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_jx(A,fun(nat,fun(A,fun(nat,A))),X),N),Y),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_3266_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jy(A,fun(nat,fun(nat,A)),X),N),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq'
tff(fact_3267_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : 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),G3),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_gy(nat,fun(nat,fun(nat,bool)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jq(fun(nat,fun(nat,A)),fun(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.triangle_reindex_eq
tff(fact_3268_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_3269_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa2) = Y )
     => ( pp(aa(product_prod(nat,nat),bool,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),X),Xa2)))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,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),X),Xa2))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_3270_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = if(A,aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3)),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3)),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),one_one(A)))),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3)) ) ).

% signed_take_bit_code
tff(fact_3271_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A6: set(nat),C3: fun(nat,A),D3: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
              & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A6)) )
           => ( groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3),A6) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C3,zero_zero(nat))),aa(nat,A,D3,zero_zero(nat))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
                & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A6)) )
           => ( groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3),A6) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_3272_or__int__rec,axiom,
    ! [K2: int,L: int] : bit_se1065995026697491101ons_or(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fdisj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ).

% or_int_rec
tff(fact_3273_execute__of__list,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_of_list(A,Xs)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ka(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Xs)),array_alloc(A,Xs,H))) ) ).

% execute_of_list
tff(fact_3274_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se4730199178511100633sh_bit(int,N,K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_3275_push__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se4730199178511100633sh_bit(int,N,K2)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% push_bit_negative_int_iff
tff(fact_3276_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,M,bit_se4730199178511100633sh_bit(A,N,A3)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),A3) ) ).

% push_bit_push_bit
tff(fact_3277_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Icc_iff
tff(fact_3278_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ico_iff
tff(fact_3279_or__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se1065995026697491101ons_or(int,K2,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_3280_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se1065995026697491101ons_or(int,K2,L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% or_negative_int_iff
tff(fact_3281_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kb(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kb(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_3282_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kc(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kc(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_3283_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kd(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kd(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = one_one(A) ) ) ) ) ) ).

% prod.delta'
tff(fact_3284_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ke(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ke(B,fun(fun(B,A),fun(B,A)),A3),B2),S) = one_one(A) ) ) ) ) ) ).

% prod.delta
tff(fact_3285_sum_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ~ pp(aa(set(B),bool,member(B,X),A6))
           => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G3,X)),groups7311177749621191930dd_sum(B,A,G3,A6)) ) ) ) ) ).

% sum.insert
tff(fact_3286_prod__pos__nat__iff,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),groups7121269368397514597t_prod(A,nat,F3,A6)))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3287_sum__of__bool__mult__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A6: set(B),P2: fun(B,bool),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kf(fun(B,bool),fun(fun(B,A),fun(B,A)),P2),F3),A6) = groups7311177749621191930dd_sum(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_3288_sum__mult__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A6: set(B),F3: fun(B,A),P2: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_kg(fun(B,A),fun(fun(B,bool),fun(B,A)),F3),P2),A6) = groups7311177749621191930dd_sum(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_3289_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A6: set(nat),C3: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
              & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A6)) )
           => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kh(fun(nat,A),fun(nat,A),C3),A6) = aa(nat,A,C3,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
                & pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A6)) )
           => ( groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kh(fun(nat,A),fun(nat,A),C3),A6) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_3290_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)),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),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_3291_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_3292_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)),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),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(A,aa(num,A,numeral_numeral(A),X),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_3293_bounded__nat__set__is__finite,axiom,
    ! [N7: set(nat),N: nat] :
      ( ! [X3: nat] :
          ( pp(aa(set(nat),bool,member(nat,X3),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),N)) )
     => pp(aa(set(nat),bool,finite_finite2(nat),N7)) ) ).

% bounded_nat_set_is_finite
tff(fact_3294_finite__nat__set__iff__bounded,axiom,
    ! [N7: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N7))
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),M3)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3295_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B2: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,N,A3)),bit_se4730199178511100633sh_bit(A,N,B2)) ) ).

% push_bit_add
tff(fact_3296_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S: set(A),S3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),bool),aTP_Lamp_ki(set(A),fun(set(A),fun(set(A),bool)),S),S3)))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3297_finite__nat__set__iff__bounded__le,axiom,
    ! [N7: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N7))
    <=> ? [M3: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),M3)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_3298_finite__M__bounded__by__nat,axiom,
    ! [P2: fun(nat,bool),I2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_kj(fun(nat,bool),fun(nat,fun(nat,bool)),P2),I2)))) ).

% finite_M_bounded_by_nat
tff(fact_3299_finite__less__ub,axiom,
    ! [F3: fun(nat,nat),U: nat] :
      ( ! [N4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),aa(nat,nat,F3,N4)))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_kk(fun(nat,nat),fun(nat,fun(nat,bool)),F3),U)))) ) ).

% finite_less_ub
tff(fact_3300_sum_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),B5: set(C),G3: fun(B,fun(C,A)),R2: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(C),bool,finite_finite2(C),B5))
           => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_km(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B5),G3),R2),A6) = groups7311177749621191930dd_sum(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_ko(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A6),G3),R2),B5) ) ) ) ) ).

% sum.swap_restrict
tff(fact_3301_prod_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B5: set(C),G3: fun(B,fun(C,A)),R2: fun(B,fun(C,bool))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(C),bool,finite_finite2(C),B5))
           => ( groups7121269368397514597t_prod(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_kp(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B5),G3),R2),A6) = groups7121269368397514597t_prod(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_kq(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A6),G3),R2),B5) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3302_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se1065995026697491101ons_or(int,X,Y))) ) ) ).

% OR_lower
tff(fact_3303_or__greater__eq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),bit_se1065995026697491101ons_or(int,K2,L))) ) ).

% or_greater_eq
tff(fact_3304_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] :
          ( ! [N4: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N4))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N4)) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = bit_se1065995026697491101ons_or(A,A3,B2) ) ) ) ).

% disjunctive_add
tff(fact_3305_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X6: set(A)] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),X6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Xa)) ) )
           => ~ pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).

% infinite_growing
tff(fact_3306_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
                & ~ ? [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),S))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa),X3)) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_3307_plus__and__or,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),bit_se5824344872417868541ns_and(int,X,Y)),bit_se1065995026697491101ons_or(int,X,Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y) ).

% plus_and_or
tff(fact_3308_sum__mono__inv,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F3: fun(I6,A),I5: set(I6),G3: fun(I6,A),I2: I6] :
          ( ( groups7311177749621191930dd_sum(I6,A,F3,I5) = groups7311177749621191930dd_sum(I6,A,G3,I5) )
         => ( ! [I3: I6] :
                ( pp(aa(set(I6),bool,member(I6,I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,I3)),aa(I6,A,G3,I3))) )
           => ( pp(aa(set(I6),bool,member(I6,I2),I5))
             => ( pp(aa(set(I6),bool,finite_finite2(I6),I5))
               => ( aa(I6,A,F3,I2) = aa(I6,A,G3,I2) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_3309_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B2))) ) ) ).

% infinite_Icc
tff(fact_3310_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B2))) ) ) ).

% infinite_Ico
tff(fact_3311_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : bit_se4730199178511100633sh_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),bit_se4730199178511100633sh_bit(A,M,A3)) ) ).

% push_bit_take_bit
tff(fact_3312_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ks(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_3313_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ku(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3314_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A),P2: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,G3,aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_kv(set(B),fun(fun(B,bool),fun(B,bool)),A6),P2))) = groups7311177749621191930dd_sum(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_kw(fun(B,A),fun(fun(B,bool),fun(B,A)),G3),P2),A6) ) ) ) ).

% sum.inter_filter
tff(fact_3315_filter__preserves__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_kx(fun(A,nat),fun(A,bool),M6))))
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ky(fun(A,nat),fun(fun(A,bool),fun(A,bool)),M6),P2)))) ) ).

% filter_preserves_multiset
tff(fact_3316_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),G3: fun(B,A),P2: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7121269368397514597t_prod(B,A,G3,aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_kv(set(B),fun(fun(B,bool),fun(B,bool)),A6),P2))) = groups7121269368397514597t_prod(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_kz(fun(B,A),fun(fun(B,bool),fun(B,A)),G3),P2),A6) ) ) ) ).

% prod.inter_filter
tff(fact_3317_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_la(A,fun(A,fun(A,bool)),A3),B2)))) ) ).

% finite_int_segment
tff(fact_3318_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
           => ( ( groups7311177749621191930dd_sum(B,A,F3,A6) = zero_zero(A) )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A6))
                 => ( aa(B,A,F3,X4) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_3319_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),T5: set(C),G3: fun(C,A),I2: fun(C,B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(C),bool,finite_finite2(C),T5))
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,member(C,X3),T5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G3,X3))) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S2))
                   => ? [Xa: C] :
                        ( pp(aa(set(C),bool,member(C,Xa),T5))
                        & ( aa(C,B,I2,Xa) = X3 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(C,A,G3,Xa))) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7311177749621191930dd_sum(B,A,F3,S2)),groups7311177749621191930dd_sum(C,A,G3,T5))) ) ) ) ) ) ).

% sum_le_included
tff(fact_3320_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [S: set(B),P2: fun(set(B),bool),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(B),bool,P2,bot_bot(set(B))))
           => ( ! [X3: B,S6: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite2(B),S6))
                 => ( ! [Y5: B] :
                        ( pp(aa(set(B),bool,member(B,Y5),S6))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,Y5)),aa(B,A,F3,X3))) )
                   => ( pp(aa(set(B),bool,P2,S6))
                     => pp(aa(set(B),bool,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X3),S6))) ) ) )
             => pp(aa(set(B),bool,P2,S)) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3321_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A6: set(I6),F3: fun(I6,A),G3: fun(I6,A)] :
          ( pp(aa(set(I6),bool,finite_finite2(I6),A6))
         => ( ! [X3: I6] :
                ( pp(aa(set(I6),bool,member(I6,X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,X3)),aa(I6,A,G3,X3))) )
           => ( ? [X5: I6] :
                  ( pp(aa(set(I6),bool,member(I6,X5),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I6,A,F3,X5)),aa(I6,A,G3,X5))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),groups7311177749621191930dd_sum(I6,A,F3,A6)),groups7311177749621191930dd_sum(I6,A,G3,A6))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3322_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),P2: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,P2,bot_bot(set(A))))
           => ( ! [B4: A,A11: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),A11))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),B4)) )
                   => ( pp(aa(set(A),bool,P2,A11))
                     => pp(aa(set(A),bool,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A11))) ) ) )
             => pp(aa(set(A),bool,P2,A6)) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3323_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),P2: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,P2,bot_bot(set(A))))
           => ( ! [B4: A,A11: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),A11))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),X5)) )
                   => ( pp(aa(set(A),bool,P2,A11))
                     => pp(aa(set(A),bool,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A11))) ) ) )
             => pp(aa(set(A),bool,P2,A6)) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3324_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R2: fun(A,fun(A,bool)),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,zero_zero(A)),zero_zero(A)))
         => ( ! [X12: A,Y12: A,X22: A,Y22: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X12),X22))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R2,Y12),Y22)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X22),Y22))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H,X3)),aa(B,A,G3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,groups7311177749621191930dd_sum(B,A,H,S)),groups7311177749621191930dd_sum(B,A,G3,S))) ) ) ) ) ) ).

% sum.related
tff(fact_3325_sum__strict__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( A6 != bot_bot(set(B)) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),groups7311177749621191930dd_sum(B,A,F3,A6)),groups7311177749621191930dd_sum(B,A,G3,A6))) ) ) ) ) ).

% sum_strict_mono
tff(fact_3326_sum_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( pp(aa(set(B),bool,member(B,X),A6))
             => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)) = groups7311177749621191930dd_sum(B,A,G3,A6) ) )
            & ( ~ pp(aa(set(B),bool,member(B,X),A6))
             => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G3,X)),groups7311177749621191930dd_sum(B,A,G3,A6)) ) ) ) ) ) ).

% sum.insert_if
tff(fact_3327_bit__push__bit__iff__int,axiom,
    ! [M: nat,K2: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_se4730199178511100633sh_bit(int,M,K2)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_3328_prod__dvd__prod__subset,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B5: set(B),A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),groups7121269368397514597t_prod(B,A,F3,A6)),groups7121269368397514597t_prod(B,A,F3,B5))) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_3329_prod__dvd__prod__subset2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [B5: set(B),A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
           => ( ! [A5: B] :
                  ( pp(aa(set(B),bool,member(B,A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F3,A5)),aa(B,A,G3,A5))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),groups7121269368397514597t_prod(B,A,F3,A6)),groups7121269368397514597t_prod(B,A,G3,B5))) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_3330_concat__bit__eq,axiom,
    ! [N: nat,K2: int,L: int] : bit_concat_bit(N,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),bit_se4730199178511100633sh_bit(int,N,L)) ).

% concat_bit_eq
tff(fact_3331_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q5: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,M,Q5)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3332_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F3: fun(B,A),B5: A,I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
           => ( ( groups7311177749621191930dd_sum(B,A,F3,S2) = B5 )
             => ( pp(aa(set(B),bool,member(B,I2),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),B5)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_3333_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
           => ( ( groups7311177749621191930dd_sum(B,A,F3,S2) = zero_zero(A) )
             => ( pp(aa(set(B),bool,member(B,I2),S2))
               => ( aa(B,A,F3,I2) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_3334_add__mset__in__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),A3: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_kx(fun(A,nat),fun(A,bool),M6))))
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_lb(fun(A,nat),fun(A,fun(A,bool)),M6),A3)))) ) ).

% add_mset_in_multiset
tff(fact_3335_sum_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A),B5: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5)) = groups7311177749621191930dd_sum(B,A,aa(set(B),fun(B,A),aTP_Lamp_lc(fun(B,A),fun(set(B),fun(B,A)),G3),B5),A6) ) ) ) ).

% sum.inter_restrict
tff(fact_3336_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_ld(fun(B,A),fun(B,bool),G3)))) = groups7311177749621191930dd_sum(B,A,G3,A6) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3337_diff__preserves__multiset,axiom,
    ! [A: $tType,M6: fun(A,nat),N7: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_kx(fun(A,nat),fun(A,bool),M6))))
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,nat),fun(A,bool),aTP_Lamp_le(fun(A,nat),fun(fun(A,nat),fun(A,bool)),M6),N7)))) ) ).

% diff_preserves_multiset
tff(fact_3338_prod_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),G3: fun(B,A),B5: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5)) = groups7121269368397514597t_prod(B,A,aa(set(B),fun(B,A),aTP_Lamp_lf(fun(B,A),fun(set(B),fun(B,A)),G3),B5),A6) ) ) ) ).

% prod.inter_restrict
tff(fact_3339_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_lg(fun(B,A),fun(B,bool),G3)))) = groups7121269368397514597t_prod(B,A,G3,A6) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3340_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_lh(nat,fun(nat,bool),M)))) ) ).

% finite_divisors_nat
tff(fact_3341_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N7),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N7)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3342_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N7),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N7)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_3343_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_li(A,fun(A,bool),A3)))) ) ).

% finite_abs_int_segment
tff(fact_3344_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),I2: B,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( pp(aa(set(B),bool,member(B,I2),I5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I2)))
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,member(B,I3),I5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),groups7311177749621191930dd_sum(B,A,F3,I5))) ) ) ) ) ) ).

% sum_pos2
tff(fact_3345_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,member(B,I3),I5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),groups7311177749621191930dd_sum(B,A,F3,I5))) ) ) ) ) ).

% sum_pos
tff(fact_3346_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( pp(aa(set(A),bool,member(A,I2),I5))
           => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F3,I2)))
             => ( ! [I3: A] :
                    ( pp(aa(set(A),bool,member(A,I3),I5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,I3))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),groups7121269368397514597t_prod(A,B,F3,I5))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3347_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( pp(aa(set(A),bool,member(A,I3),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F3,I3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),groups7121269368397514597t_prod(A,B,F3,I5))) ) ) ) ) ).

% less_1_prod
tff(fact_3348_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T8: set(B),S: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( groups7311177749621191930dd_sum(B,A,G3,T8) = groups7311177749621191930dd_sum(B,A,H,S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_3349_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T8: set(B),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,H,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( groups7311177749621191930dd_sum(B,A,G3,S) = groups7311177749621191930dd_sum(B,A,H,T8) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_3350_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T8: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( groups7311177749621191930dd_sum(B,A,G3,T8) = groups7311177749621191930dd_sum(B,A,G3,S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_3351_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T8: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( groups7311177749621191930dd_sum(B,A,G3,S) = groups7311177749621191930dd_sum(B,A,G3,T8) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_3352_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C4: set(B),A6: set(B),B5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C4))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,member(B,A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A6)))
                   => ( aa(B,A,G3,A5) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,member(B,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B5)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( groups7311177749621191930dd_sum(B,A,G3,C4) = groups7311177749621191930dd_sum(B,A,H,C4) )
                   => ( groups7311177749621191930dd_sum(B,A,G3,A6) = groups7311177749621191930dd_sum(B,A,H,B5) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_3353_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C4: set(B),A6: set(B),B5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C4))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,member(B,A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A6)))
                   => ( aa(B,A,G3,A5) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,member(B,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B5)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( groups7311177749621191930dd_sum(B,A,G3,A6) = groups7311177749621191930dd_sum(B,A,H,B5) )
                  <=> ( groups7311177749621191930dd_sum(B,A,G3,C4) = groups7311177749621191930dd_sum(B,A,H,C4) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_3354_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: set(B),A6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),A6))
           => ( groups7311177749621191930dd_sum(B,A,G3,A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5))),groups7311177749621191930dd_sum(B,A,G3,B5)) ) ) ) ) ).

% sum.subset_diff
tff(fact_3355_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A6: set(B),B5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A6))
           => ( groups7311177749621191930dd_sum(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),groups7311177749621191930dd_sum(B,A,F3,A6)),groups7311177749621191930dd_sum(B,A,F3,B5)) ) ) ) ) ).

% sum_diff
tff(fact_3356_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B5: set(B),A6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),A6))
           => ( groups7121269368397514597t_prod(B,A,G3,A6) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5))),groups7121269368397514597t_prod(B,A,G3,B5)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3357_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T8: set(B),S: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( groups7121269368397514597t_prod(B,A,G3,T8) = groups7121269368397514597t_prod(B,A,H,S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3358_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T8: set(B),S: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,H,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),S))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( groups7121269368397514597t_prod(B,A,G3,S) = groups7121269368397514597t_prod(B,A,H,T8) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3359_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T8: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G3,T8) = groups7121269368397514597t_prod(B,A,G3,S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3360_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T8: set(B),S: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T8))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G3,S) = groups7121269368397514597t_prod(B,A,G3,T8) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3361_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C4: set(B),A6: set(B),B5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C4))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,member(B,A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A6)))
                   => ( aa(B,A,G3,A5) = one_one(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,member(B,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B5)))
                     => ( aa(B,A,H,B4) = one_one(A) ) )
                 => ( ( groups7121269368397514597t_prod(B,A,G3,C4) = groups7121269368397514597t_prod(B,A,H,C4) )
                   => ( groups7121269368397514597t_prod(B,A,G3,A6) = groups7121269368397514597t_prod(B,A,H,B5) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3362_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C4: set(B),A6: set(B),B5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),C4))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C4))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,member(B,A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),A6)))
                   => ( aa(B,A,G3,A5) = one_one(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,member(B,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C4),B5)))
                     => ( aa(B,A,H,B4) = one_one(A) ) )
                 => ( ( groups7121269368397514597t_prod(B,A,G3,A6) = groups7121269368397514597t_prod(B,A,H,B5) )
                  <=> ( groups7121269368397514597t_prod(B,A,G3,C4) = groups7121269368397514597t_prod(B,A,H,C4) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3363_sum_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,A6)),groups7311177749621191930dd_sum(B,A,G3,B5)) ) ) ) ) ).

% sum.union_inter
tff(fact_3364_sum_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A),B5: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,G3,A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5))) ) ) ) ).

% sum.Int_Diff
tff(fact_3365_prod_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))),groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G3,A6)),groups7121269368397514597t_prod(B,A,G3,B5)) ) ) ) ) ).

% prod.union_inter
tff(fact_3366_sum__diff__nat,axiom,
    ! [A: $tType,B5: set(A),A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
       => ( groups7311177749621191930dd_sum(A,nat,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),groups7311177749621191930dd_sum(A,nat,F3,A6)),groups7311177749621191930dd_sum(A,nat,F3,B5)) ) ) ) ).

% sum_diff_nat
tff(fact_3367_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] : bit_se4730199178511100633sh_bit(A,M,bit_se2239418461657761734s_mask(A,N)) = bit_se5824344872417868541ns_and(A,bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),bit_ri4277139882892585799ns_not(A,bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_3368_sum_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),P2: fun(B,bool),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lj(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P2),H),G3),A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2)))),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P2))))) ) ) ) ).

% sum.If_cases
tff(fact_3369_prod_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),P2: fun(B,bool),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lk(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P2),H),G3),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2)))),groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P2))))) ) ) ) ).

% prod.If_cases
tff(fact_3370_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),A6))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) ) )
           => ( ( A6 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),groups7121269368397514597t_prod(B,A,F3,A6)),groups7121269368397514597t_prod(B,A,G3,A6))) ) ) ) ) ).

% prod_mono_strict
tff(fact_3371_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B5: set(B),A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
           => ( ! [B4: B] :
                  ( pp(aa(set(B),bool,member(B,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A6)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,B4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7311177749621191930dd_sum(B,A,F3,A6)),groups7311177749621191930dd_sum(B,A,F3,B5))) ) ) ) ) ).

% sum_mono2
tff(fact_3372_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,A6)),groups7311177749621191930dd_sum(B,A,G3,B5)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_3373_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G3,X)),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3374_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,member(B,X),A6))
           => ( groups7311177749621191930dd_sum(B,A,G3,A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G3,X)),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_3375_sum__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A6: set(B),B5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( groups7311177749621191930dd_sum(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,F3,A6)),groups7311177749621191930dd_sum(B,A,F3,B5))),groups7311177749621191930dd_sum(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))) ) ) ) ) ).

% sum_Un
tff(fact_3376_sum_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5) = bot_bot(set(B)) )
             => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,A6)),groups7311177749621191930dd_sum(B,A,G3,B5)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_3377_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G3,A6)),groups7121269368397514597t_prod(B,A,G3,B5)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_3378_prod_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5) = bot_bot(set(B)) )
             => ( groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G3,A6)),groups7121269368397514597t_prod(B,A,G3,B5)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_3379_sum_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5))),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A6)))),groups7311177749621191930dd_sum(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))) ) ) ) ) ).

% sum.union_diff2
tff(fact_3380_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A6: set(A),B5: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))
         => ( groups7311177749621191930dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),groups7311177749621191930dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5))),groups7311177749621191930dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)))),groups7311177749621191930dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5))) ) ) ) ).

% sum_Un2
tff(fact_3381_prod_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5))),groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A6)))),groups7121269368397514597t_prod(B,A,G3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))) ) ) ) ) ).

% prod.union_diff2
tff(fact_3382_sum__Un__nat,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( groups7311177749621191930dd_sum(A,nat,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups7311177749621191930dd_sum(A,nat,F3,A6)),groups7311177749621191930dd_sum(A,nat,F3,B5))),groups7311177749621191930dd_sum(A,nat,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5))) ) ) ) ).

% sum_Un_nat
tff(fact_3383_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),A3: B,B2: fun(B,A),C3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ll(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3),S) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A3)),groups7311177749621191930dd_sum(B,A,C3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ll(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3),S) = groups7311177749621191930dd_sum(B,A,C3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_3384_sum__div__partition,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A6: set(B),F3: fun(B,A),B2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7311177749621191930dd_sum(B,A,F3,A6)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_lm(fun(B,A),fun(A,fun(B,A)),F3),B2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_ln(fun(B,A),fun(A,fun(B,bool)),F3),B2))))),aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7311177749621191930dd_sum(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_lo(fun(B,A),fun(A,fun(B,bool)),F3),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_3385_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A),C3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lp(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),groups7121269368397514597t_prod(B,A,C3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lp(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3),S) = groups7121269368397514597t_prod(B,A,C3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_3386_effect__of__listI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H5: heap_ext(product_unit),Xs: list(A),H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H5) = array_alloc(A,Xs,H) )
         => heap_Time_effect(array(A),array_of_list(A,Xs),H,H5,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% effect_of_listI
tff(fact_3387_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B5: set(A),A6: set(A),B2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B5))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
           => ( pp(aa(set(A),bool,member(A,B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)))
             => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F3,B2)))
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,member(A,X3),B5))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3))) )
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),groups7311177749621191930dd_sum(A,B,F3,A6)),groups7311177749621191930dd_sum(A,B,F3,B5))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3388_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I2: C,A6: set(C),F3: fun(C,B)] :
          ( pp(aa(set(C),bool,member(C,I2),A6))
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,member(C,X3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A6),aa(set(C),set(C),aa(C,fun(set(C),set(C)),insert(C),I2),bot_bot(set(C))))))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F3,X3))) )
           => ( pp(aa(set(C),bool,finite_finite2(C),A6))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F3,I2)),groups7311177749621191930dd_sum(C,B,F3,A6))) ) ) ) ) ).

% member_le_sum
tff(fact_3389_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B5: set(A),A6: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B5))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,member(A,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,B4))) )
             => ( ! [A5: A] :
                    ( pp(aa(set(A),bool,member(A,A5),A6))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,A5))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F3,A6)),groups7121269368397514597t_prod(A,B,F3,B5))) ) ) ) ) ) ).

% prod_mono2
tff(fact_3390_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A6: set(B),B5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B5))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5)))
                 => ( aa(B,A,F3,X3) != zero_zero(A) ) )
             => ( groups7121269368397514597t_prod(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,F3,A6)),groups7121269368397514597t_prod(B,A,F3,B5))),groups7121269368397514597t_prod(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))) ) ) ) ) ) ).

% prod_Un
tff(fact_3391_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se1065995026697491101ons_or(A,A3,one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))) ) ).

% or_one_eq
tff(fact_3392_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se1065995026697491101ons_or(A,one_one(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),A3))) ) ).

% one_or_eq
tff(fact_3393_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se1065995026697491101ons_or(int,X,Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),N))) ) ) ) ).

% OR_upper
tff(fact_3394_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : bit_se1065995026697491101ons_or(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,M)),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(int,aa(num,int,numeral_numeral(int),M),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_3395_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lq(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_3396_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : bit_se1065995026697491101ons_or(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,M)),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(int,aa(num,int,numeral_numeral(int),M),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_3397_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : bit_se1065995026697491101ons_or(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,M)),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(int,aa(num,int,numeral_numeral(int),M),bit_ri4277139882892585799ns_not(int,aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_3398_of__list__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : array_of_list(A,Xs) = heap_Time_heap(array(A),aTP_Lamp_lr(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).

% of_list_def
tff(fact_3399_finite__Collect__le__nat,axiom,
    ! [K2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ck(nat,fun(nat,bool)),K2)))) ).

% finite_Collect_le_nat
tff(fact_3400_finite__Collect__less__nat,axiom,
    ! [K2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),K2)))) ).

% finite_Collect_less_nat
tff(fact_3401_finite__Collect__subsets,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ls(set(A),fun(set(A),bool),A6)))) ) ).

% finite_Collect_subsets
tff(fact_3402_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P2: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(A),bool,P2,bot_bot(set(A))))
       => ( ! [T9: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T9),S))
             => ( pp(aa(set(A),bool,P2,T9))
               => ? [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T9)))
                    & pp(aa(set(A),bool,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),T9))) ) ) )
         => pp(aa(set(A),bool,P2,S)) ) ) ) ).

% finite_induct_select
tff(fact_3403_finite__Un,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G5)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),F4))
        & pp(aa(set(A),bool,finite_finite2(A),G5)) ) ) ).

% finite_Un
tff(fact_3404_finite__Collect__conjI,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
        | pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),Q2))) )
     => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ah(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2)))) ) ).

% finite_Collect_conjI
tff(fact_3405_finite__Collect__disjI,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ai(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2))))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
        & pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),Q2))) ) ) ).

% finite_Collect_disjI
tff(fact_3406_finite__interval__int1,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_lt(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int1
tff(fact_3407_finite__interval__int4,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_lu(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int4
tff(fact_3408_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,L,U))) ).

% finite_atLeastLessThan_integer
tff(fact_3409_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or1337092689740270186AtMost(code_integer,L,U))) ).

% finite_atLeastAtMost_integer
tff(fact_3410_finite__interval__int2,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_lv(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int2
tff(fact_3411_finite__interval__int3,axiom,
    ! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_lw(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int3
tff(fact_3412_finite__maxlen,axiom,
    ! [A: $tType,M6: set(list(A))] :
      ( pp(aa(set(list(A)),bool,finite_finite2(list(A)),M6))
     => ? [N4: nat] :
        ! [X5: list(A)] :
          ( pp(aa(set(list(A)),bool,member(list(A),X5),M6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X5)),N4)) ) ) ).

% finite_maxlen
tff(fact_3413_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero(int) )
     => pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aTP_Lamp_lx(int,fun(int,bool),I2)))) ) ).

% finite_divisors_int
tff(fact_3414_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U))) ).

% finite_atLeastZeroLessThan_integer
tff(fact_3415_not__finite__existsD,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
     => ? [X_1: A] : pp(aa(A,bool,P2,X_1)) ) ).

% not_finite_existsD
tff(fact_3416_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B),R2: fun(A,fun(B,bool))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A6))
             => ? [Xa: B] :
                  ( pp(aa(set(B),bool,member(B,Xa),B5))
                  & pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Xa)) ) )
         => ? [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),B5))
              & ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ly(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),A6),R2),X3)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3417_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X3))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3418_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3419_rev__finite__subset,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
       => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% rev_finite_subset
tff(fact_3420_infinite__super,axiom,
    ! [A: $tType,S: set(A),T8: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T8))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
       => ~ pp(aa(set(A),bool,finite_finite2(A),T8)) ) ) ).

% infinite_super
tff(fact_3421_finite__subset,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% finite_subset
tff(fact_3422_finite__UnI,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,finite_finite2(A),G5))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G5))) ) ) ).

% finite_UnI
tff(fact_3423_Un__infinite,axiom,
    ! [A: $tType,S: set(A),T8: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
     => ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T8))) ) ).

% Un_infinite
tff(fact_3424_infinite__Un,axiom,
    ! [A: $tType,S: set(A),T8: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T8)))
    <=> ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
        | ~ pp(aa(set(A),bool,finite_finite2(A),T8)) ) ) ).

% infinite_Un
tff(fact_3425_finite__psubset__induct,axiom,
    ! [A: $tType,A6: set(A),P2: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ! [A11: set(A)] :
            ( pp(aa(set(A),bool,finite_finite2(A),A11))
           => ( ! [B9: set(A)] :
                  ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B9),A11))
                 => pp(aa(set(A),bool,P2,B9)) )
             => pp(aa(set(A),bool,P2,A11)) ) )
       => pp(aa(set(A),bool,P2,A6)) ) ) ).

% finite_psubset_induct
tff(fact_3426_Suc__0__or__eq,axiom,
    ! [N: nat] : bit_se1065995026697491101ons_or(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_3427_or__Suc__0__eq,axiom,
    ! [N: nat] : bit_se1065995026697491101ons_or(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_3428_or__nat__rec,axiom,
    ! [M: nat,N: nat] : bit_se1065995026697491101ons_or(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),M)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))) ).

% or_nat_rec
tff(fact_3429_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A6))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3430_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A6))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X3))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3431_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A6: set(A),P2: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F4),A6))
       => ( pp(aa(set(A),bool,P2,bot_bot(set(A))))
         => ( ! [A5: A,F7: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F7))
               => ( pp(aa(set(A),bool,member(A,A5),A6))
                 => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F7),A6))
                   => ( ~ pp(aa(set(A),bool,member(A,A5),F7))
                     => ( pp(aa(set(A),bool,P2,F7))
                       => pp(aa(set(A),bool,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),F7))) ) ) ) ) )
           => pp(aa(set(A),bool,P2,F4)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3432_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A6: set(A),P2: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),F4))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F4),A6))
       => ( pp(aa(set(A),bool,P2,bot_bot(set(A))))
         => ( ! [A5: A,F7: set(A)] :
                ( pp(aa(set(A),bool,finite_finite2(A),F7))
               => ( pp(aa(set(A),bool,member(A,A5),A6))
                 => ( ~ pp(aa(set(A),bool,member(A,A5),F7))
                   => ( pp(aa(set(A),bool,P2,F7))
                     => pp(aa(set(A),bool,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),F7))) ) ) ) )
           => pp(aa(set(A),bool,P2,F4)) ) ) ) ) ).

% finite_subset_induct
tff(fact_3433_finite__remove__induct,axiom,
    ! [A: $tType,B5: set(A),P2: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,P2,bot_bot(set(A))))
       => ( ! [A11: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A11))
             => ( ( A11 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),B5))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(set(A),bool,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A11),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P2,A11)) ) ) ) )
         => pp(aa(set(A),bool,P2,B5)) ) ) ) ).

% finite_remove_induct
tff(fact_3434_remove__induct,axiom,
    ! [A: $tType,P2: fun(set(A),bool),B5: set(A)] :
      ( pp(aa(set(A),bool,P2,bot_bot(set(A))))
     => ( ( ~ pp(aa(set(A),bool,finite_finite2(A),B5))
         => pp(aa(set(A),bool,P2,B5)) )
       => ( ! [A11: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),A11))
             => ( ( A11 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),B5))
                 => ( ! [X5: A] :
                        ( pp(aa(set(A),bool,member(A,X5),A11))
                       => pp(aa(set(A),bool,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A11),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P2,A11)) ) ) ) )
         => pp(aa(set(A),bool,P2,B5)) ) ) ) ).

% remove_induct
tff(fact_3435_set__encode__insert,axiom,
    ! [A6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
     => ( ~ pp(aa(set(nat),bool,member(nat,N),A6))
       => ( nat_set_encode(aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N),A6)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),N)),nat_set_encode(A6)) ) ) ) ).

% set_encode_insert
tff(fact_3436_finite__nat__bounded,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => ? [K: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ).

% finite_nat_bounded
tff(fact_3437_finite__nat__iff__bounded,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_lessThan(nat),K3))) ) ).

% finite_nat_iff_bounded
tff(fact_3438_finite__nat__iff__bounded__le,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_atMost(nat),K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_3439_in__finite__psubset,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A6),B5)),finite_psubset(A)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
        & pp(aa(set(A),bool,finite_finite2(A),B5)) ) ) ).

% in_finite_psubset
tff(fact_3440_infinite__int__iff__unbounded__le,axiom,
    ! [S: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M3),aa(int,int,abs_abs(int),N3)))
          & pp(aa(set(int),bool,member(int,N3),S)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_3441_infinite__int__iff__unbounded,axiom,
    ! [S: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ! [M3: int] :
        ? [N3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M3),aa(int,int,abs_abs(int),N3)))
          & pp(aa(set(int),bool,member(int,N3),S)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_3442_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_lz(set(A),fun(set(A),bool)))) ).

% finite_psubset_def
tff(fact_3443_unbounded__k__infinite,axiom,
    ! [K2: nat,S: set(nat)] :
      ( ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),M5))
         => ? [N8: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N8))
              & pp(aa(set(nat),bool,member(nat,N8),S)) ) )
     => ~ pp(aa(set(nat),bool,finite_finite2(nat),S)) ) ).

% unbounded_k_infinite
tff(fact_3444_infinite__nat__iff__unbounded,axiom,
    ! [S: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N3))
          & pp(aa(set(nat),bool,member(nat,N3),S)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_3445_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S))
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N3))
          & pp(aa(set(nat),bool,member(nat,N3),S)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_3446_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
          | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( bit_se1065995026697491101ons_or(int,K2,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) )
      & ( ~ ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
            | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( ( ( K2 = zero_zero(int) )
           => ( bit_se1065995026697491101ons_or(int,K2,L) = L ) )
          & ( ( K2 != zero_zero(int) )
           => ( ( ( L = zero_zero(int) )
               => ( bit_se1065995026697491101ons_or(int,K2,L) = K2 ) )
              & ( ( L != zero_zero(int) )
               => ( bit_se1065995026697491101ons_or(int,K2,L) = 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,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_3447_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( bit_se1065995026697491101ons_or(nat,M,N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( bit_se1065995026697491101ons_or(nat,M,N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_se1065995026697491101ons_or(nat,M,N) = 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),aa(num,num,bit0,one)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),bit_se1065995026697491101ons_or(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_3448_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F3: fun(A,B),I2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),F4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(A,bool)),I5),F3))),F4))
           => ( ( pp(aa(set(A),bool,member(A,I2),I5))
               => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I5)),aa(A,B,F3,I2)) ) )
              & ( ~ pp(aa(set(A),bool,member(A,I2),I5))
               => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F3,I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3449_even__sum__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),groups7311177749621191930dd_sum(B,A,F3,A6)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(set(B),nat,finite_card(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_mb(set(B),fun(fun(B,A),fun(B,bool)),A6),F3))))) ) ) ) ).

% even_sum_iff
tff(fact_3450_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one )
           => ( ? [N4: num] : Xa2 = aa(num,num,bit0,N4)
             => ( Y != aa(num,option(num),some(num),one) ) ) )
         => ( ( ( X = one )
             => ( ? [N4: num] : Xa2 = aa(num,num,bit1,N4)
               => ( Y != none(num) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa(num,num,bit0,M5) )
                 => ( ( Xa2 = one )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N4: num] :
                        ( ( Xa2 = aa(num,num,bit0,N4) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N4)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N4: num] :
                          ( ( Xa2 = aa(num,num,bit1,N4) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N4)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit1,M5) )
                       => ( ( Xa2 = one )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N4: num] :
                              ( ( Xa2 = aa(num,num,bit0,N4) )
                             => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one),aTP_Lamp_es(num,option(num)),bit_and_not_num(M5,N4)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N4: num] :
                                ( ( Xa2 = aa(num,num,bit1,N4) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_3451_card__Collect__less__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N))) = N ).

% card_Collect_less_nat
tff(fact_3452_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% max.bounded_iff
tff(fact_3453_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_3454_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb1
tff(fact_3455_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z4))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z4)) ) ) ) ).

% max_less_iff_conj
tff(fact_3456_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_3457_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb3
tff(fact_3458_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).

% max_bot2
tff(fact_3459_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).

% max_bot
tff(fact_3460_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xo: option(B),Y: A] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),Xo) = aa(A,option(A),some(A),Y) )
    <=> ? [Z2: B] :
          ( ( Xo = aa(B,option(B),some(B),Z2) )
          & ( aa(B,A,F3,Z2) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_3461_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: option(B)] :
      ( ( none(A) = aa(option(B),option(A),map_option(B,A,F3),X) )
    <=> ( X = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_3462_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_3463_option_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: option(A)] :
      ( ( aa(option(A),option(B),map_option(A,B,F3),A3) = none(B) )
    <=> ( A3 = none(A) ) ) ).

% option.map_disc_iff
tff(fact_3464_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).

% max_0R
tff(fact_3465_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).

% max_0L
tff(fact_3466_max__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),zero_zero(nat)) = A3 ).

% max_nat.right_neutral
tff(fact_3467_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_3468_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A3) = A3 ).

% max_nat.left_neutral
tff(fact_3469_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_3470_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)) ).

% max_Suc_Suc
tff(fact_3471_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ck(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_3472_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_3473_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: A,H: fun(B,A),F3: fun(C,B),X: option(C)] : case_option(A,B,G3,H,aa(option(C),option(B),map_option(C,B,F3),X)) = case_option(A,C,G3,aa(fun(C,B),fun(C,A),comp(B,A,C,H),F3),X) ).

% case_map_option
tff(fact_3474_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A6: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_mc(A,fun(B,A),Y),A6) = aa(nat,A,power_power(A,Y),aa(set(B),nat,finite_card(B),A6)) ) ).

% prod_constant
tff(fact_3475_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_3476_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_3477_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_3478_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A6: set(B)] : groups7311177749621191930dd_sum(B,A,aTP_Lamp_md(A,fun(B,A),Y),A6) = 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),A6))),Y) ) ).

% sum_constant
tff(fact_3479_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_3480_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
         => ( ( pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1027152243600224163dd_sum(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = groups1027152243600224163dd_sum(B,A,P3,I5) ) )
            & ( ~ pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1027152243600224163dd_sum(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P3,I2)),groups1027152243600224163dd_sum(B,A,P3,I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_3481_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)) )
    <=> ( A3 != B2 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_3482_sum__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A6: set(B),P2: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),A6))
           => ( groups7311177749621191930dd_sum(B,A,aTP_Lamp_me(fun(B,bool),fun(B,A),P2),A6) = aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_3483_option_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X2: A] : aa(option(A),option(B),map_option(A,B,F3),aa(A,option(A),some(A),X2)) = aa(B,option(B),some(B),aa(A,B,F3,X2)) ).

% option.simps(9)
tff(fact_3484_map__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y: option(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( X = Y )
     => ( ! [A5: A] :
            ( ( Y = aa(A,option(A),some(A),A5) )
           => ( aa(A,B,F3,A5) = aa(A,B,G3,A5) ) )
       => ( aa(option(A),option(B),map_option(A,B,F3),X) = aa(option(A),option(B),map_option(A,B,G3),Y) ) ) ) ).

% map_option_cong
tff(fact_3485_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_3486_map__option_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C),G3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),comp(option(B),option(C),option(A),map_option(B,C,F3)),map_option(A,B,G3)) = map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G3)) ).

% map_option.comp
tff(fact_3487_option_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G3: fun(B,C),F3: fun(A,B),V2: option(A)] : aa(option(B),option(C),map_option(B,C,G3),aa(option(A),option(B),map_option(A,B,F3),V2)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G3),F3)),V2) ).

% option.map_comp
tff(fact_3488_map__option_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(B,C),G3: fun(A,B),Option: option(A)] : aa(option(B),option(C),map_option(B,C,F3),aa(option(A),option(B),map_option(A,B,G3),Option)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G3)),Option) ).

% map_option.compositionality
tff(fact_3489_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) ).

% nat_mult_max_left
tff(fact_3490_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)) ).

% nat_mult_max_right
tff(fact_3491_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.strict_coboundedI2
tff(fact_3492_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.strict_coboundedI1
tff(fact_3493_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_3494_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% max.strict_boundedE
tff(fact_3495_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z4: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),Y)) ) ) ) ).

% less_max_iff_disj
tff(fact_3496_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q5)) ).

% nat_add_max_left
tff(fact_3497_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)) ).

% nat_add_max_right
tff(fact_3498_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_3499_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_3500_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_3501_option_Omap__ident,axiom,
    ! [A: $tType,T5: option(A)] : aa(option(A),option(A),map_option(A,A,aTP_Lamp_cc(A,A)),T5) = T5 ).

% option.map_ident
tff(fact_3502_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI2
tff(fact_3503_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI1
tff(fact_3504_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_3505_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_3506_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z4: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),Y)) ) ) ) ).

% le_max_iff_disj
tff(fact_3507_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded2
tff(fact_3508_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded1
tff(fact_3509_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.order_iff
tff(fact_3510_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3)) ) ) ) ).

% max.boundedI
tff(fact_3511_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% max.boundedE
tff(fact_3512_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% max.orderI
tff(fact_3513_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.orderE
tff(fact_3514_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,D3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ) ).

% max.mono
tff(fact_3515_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_3516_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_3517_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ) ).

% max_def
tff(fact_3518_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z4)) ) ).

% max_add_distrib_right
tff(fact_3519_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z4) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z4)) ) ).

% max_add_distrib_left
tff(fact_3520_option_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B)] : aa(option(A),option(B),map_option(A,B,F3),none(A)) = none(B) ).

% option.simps(8)
tff(fact_3521_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa) = Xa ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa) = X5 ) ) ) ) ).

% max_def_raw
tff(fact_3522_n__subsets,axiom,
    ! [A: $tType,A6: set(A),K2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(nat,fun(set(A),bool),aTP_Lamp_mf(set(A),fun(nat,fun(set(A),bool)),A6),K2))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A6)),K2) ) ) ).

% n_subsets
tff(fact_3523_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G3,aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_mg(fun(B,A),fun(set(B),fun(B,bool)),G3),I5))) = groups1027152243600224163dd_sum(B,A,G3,I5) ) ).

% sum.non_neutral'
tff(fact_3524_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,nat),G3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),comp(option(B),nat,option(A),size_option(B,F3)),map_option(A,B,G3)) = size_option(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F3),G3)) ).

% option.size_gen_o_map
tff(fact_3525_card__subset__eq,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
       => ( ( aa(set(A),nat,finite_card(A),A6) = aa(set(A),nat,finite_card(A),B5) )
         => ( A6 = B5 ) ) ) ) ).

% card_subset_eq
tff(fact_3526_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ? [B8: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = N )
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),A6)) ) ) ).

% infinite_arbitrarily_large
tff(fact_3527_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),M) ).

% nat_minus_add_max
tff(fact_3528_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B5: set(A),A6: set(B),R3: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( ! [A5: B] :
            ( pp(aa(set(B),bool,member(B,A5),A6))
           => ? [B10: A] :
                ( pp(aa(set(A),bool,member(A,B10),B5))
                & pp(aa(A,bool,aa(B,fun(A,bool),R3,A5),B10)) ) )
       => ( ! [A12: B,A23: B,B4: A] :
              ( pp(aa(set(B),bool,member(B,A12),A6))
             => ( pp(aa(set(B),bool,member(B,A23),A6))
               => ( pp(aa(set(A),bool,member(A,B4),B5))
                 => ( pp(aa(A,bool,aa(B,fun(A,bool),R3,A12),B4))
                   => ( pp(aa(A,bool,aa(B,fun(A,bool),R3,A23),B4))
                     => ( A12 = A23 ) ) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A6)),aa(set(A),nat,finite_card(A),B5))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3529_card__insert__le,axiom,
    ! [A: $tType,A6: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)))) ).

% card_insert_le
tff(fact_3530_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T5: set(B),R2: fun(A,fun(B,bool)),K2: fun(B,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => ( pp(aa(set(B),bool,finite_finite2(B),T5))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),T5))
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ly(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S2),R2),X3))) = aa(B,nat,K2,X3) ) )
         => ( groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_mi(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T5),R2),S2) = groups7311177749621191930dd_sum(B,nat,K2,T5) ) ) ) ) ).

% sum_multicount_gen
tff(fact_3531_card__eq__sum,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),nat,finite_card(A),A6) = groups7311177749621191930dd_sum(A,nat,aTP_Lamp_mj(A,nat),A6) ).

% card_eq_sum
tff(fact_3532_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_mk(nat,fun(nat,nat),N),M) ).

% max_Suc2
tff(fact_3533_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_ml(nat,fun(nat,nat),N),M) ).

% max_Suc1
tff(fact_3534_sum_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fz(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G3,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3535_card__ge__0__finite,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A6)))
     => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ).

% card_ge_0_finite
tff(fact_3536_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C4: nat] :
      ( ! [G6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),G6),F4))
         => ( pp(aa(set(A),bool,finite_finite2(A),G6))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G6)),C4)) ) )
     => ( pp(aa(set(A),bool,finite_finite2(A),F4))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C4)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3537_card__seteq,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B5)),aa(set(A),nat,finite_card(A),A6)))
         => ( A6 = B5 ) ) ) ) ).

% card_seteq
tff(fact_3538_card__mono,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5))) ) ) ).

% card_mono
tff(fact_3539_obtain__subset__with__card__n,axiom,
    ! [A: $tType,N: nat,S: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),S)))
     => ~ ! [T9: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),S))
           => ( ( aa(set(A),nat,finite_card(A),T9) = N )
             => ~ pp(aa(set(A),bool,finite_finite2(A),T9)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3540_card__less__sym__Diff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3541_card__le__sym__Diff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3542_psubset__card__mono,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5))) ) ) ).

% psubset_card_mono
tff(fact_3543_card__Un__le,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),A6),B5))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)))) ).

% card_Un_le
tff(fact_3544_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ck(nat,fun(nat,bool)),aTP_Lamp_cl(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_3545_card__less__Suc2,axiom,
    ! [M6: set(nat),I2: nat] :
      ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_mm(set(nat),fun(nat,fun(nat,bool)),M6),I2))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_mn(set(nat),fun(nat,fun(nat,bool)),M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_3546_card__less__Suc,axiom,
    ! [M6: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_mm(set(nat),fun(nat,fun(nat,bool)),M6),I2)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_mn(set(nat),fun(nat,fun(nat,bool)),M6),I2))) ) ) ).

% card_less_Suc
tff(fact_3547_card__less,axiom,
    ! [M6: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_mn(set(nat),fun(nat,fun(nat,bool)),M6),I2))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3548_subset__card__intvl__is__intvl,axiom,
    ! [A6: set(nat),K2: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),A6),set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A6)))))
     => ( A6 = set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A6))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_3549_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_mo(A,fun(A,bool)),aTP_Lamp_mp(A,fun(A,bool))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_3550_sum__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),A6: set(A)] : groups7311177749621191930dd_sum(A,nat,aTP_Lamp_mq(fun(A,nat),fun(A,nat),F3),A6) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups7311177749621191930dd_sum(A,nat,F3,A6)),aa(set(A),nat,finite_card(A),A6)) ).

% sum_Suc
tff(fact_3551_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S: set(A),T8: set(B),R2: fun(A,fun(B,bool)),K2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(B),bool,finite_finite2(B),T8))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),T8))
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ly(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S),R2),X3))) = K2 ) )
         => ( groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_mi(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T8),R2),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(set(B),nat,finite_card(B),T8)) ) ) ) ) ).

% sum_multicount
tff(fact_3552_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T8: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G3,S) = groups1027152243600224163dd_sum(B,A,G3,T8) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3553_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T8: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G3,T8) = groups1027152243600224163dd_sum(B,A,G3,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3554_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T8: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,H,I3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G3,S) = groups1027152243600224163dd_sum(B,A,H,T8) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3555_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T8: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G3,T8) = groups1027152243600224163dd_sum(B,A,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3556_map__option__case,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Y: option(B)] : aa(option(B),option(A),map_option(B,A,F3),Y) = case_option(option(A),B,none(A),aTP_Lamp_mr(fun(B,A),fun(B,option(A)),F3),Y) ).

% map_option_case
tff(fact_3557_card__gt__0__iff,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A6)))
    <=> ( ( A6 != bot_bot(set(A)) )
        & pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% card_gt_0_iff
tff(fact_3558_sum__bounded__above,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A6: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),K5)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7311177749621191930dd_sum(B,A,F3,A6)),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),A6))),K5))) ) ) ).

% sum_bounded_above
tff(fact_3559_sum__bounded__below,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A6: set(B),K5: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K5),aa(B,A,F3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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),A6))),K5)),groups7311177749621191930dd_sum(B,A,F3,A6))) ) ) ).

% sum_bounded_below
tff(fact_3560_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(nat,nat,suc,zero_zero(nat))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),A6))
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3561_card__le__Suc__iff,axiom,
    ! [A: $tType,N: nat,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),A6)))
    <=> ? [A9: A,B11: set(A)] :
          ( ( A6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A9),B11) )
          & ~ pp(aa(set(A),bool,member(A,A9),B11))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),B11)))
          & pp(aa(set(A),bool,finite_finite2(A),B11)) ) ) ).

% card_le_Suc_iff
tff(fact_3562_card__1__singletonI,axiom,
    ! [A: $tType,S: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( ( aa(set(A),nat,finite_card(A),S) = one_one(nat) )
       => ( pp(aa(set(A),bool,member(A,X),S))
         => ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_3563_card__Diff1__le,axiom,
    ! [A: $tType,A6: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A6))) ).

% card_Diff1_le
tff(fact_3564_card__Diff__subset,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ).

% card_Diff_subset
tff(fact_3565_card__psubset,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5)) ) ) ) ).

% card_psubset
tff(fact_3566_card__Un__Int,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(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)),A6),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5))) ) ) ) ).

% card_Un_Int
tff(fact_3567_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)))) ) ).

% diff_card_le_card_Diff
tff(fact_3568_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),G3))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fz(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G3,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3569_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P3: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1027152243600224163dd_sum(B,A,P3,I5) = groups7311177749621191930dd_sum(B,A,P3,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1027152243600224163dd_sum(B,A,P3,I5) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_3570_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N7),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N7)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_3571_card__sum__le__nat__sum,axiom,
    ! [S: set(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fn(nat,nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S)))),groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fn(nat,nat),S))) ).

% card_sum_le_nat_sum
tff(fact_3572_card__insert__disjoint_H,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ~ pp(aa(set(A),bool,member(A,X),A6))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6))),aa(nat,nat,suc,zero_zero(nat))) = aa(set(A),nat,finite_card(A),A6) ) ) ) ).

% card_insert_disjoint'
tff(fact_3573_card__Diff1__less__iff,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A6)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A6))
        & pp(aa(set(A),bool,member(A,X),A6)) ) ) ).

% card_Diff1_less_iff
tff(fact_3574_card__Diff2__less,axiom,
    ! [A: $tType,A6: set(A),X: A,Y: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,member(A,X),A6))
       => ( pp(aa(set(A),bool,member(A,Y),A6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A6))) ) ) ) ).

% card_Diff2_less
tff(fact_3575_card__Diff1__less,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,member(A,X),A6))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A6))) ) ) ).

% card_Diff1_less
tff(fact_3576_card__Un__disjoint,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = 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)),A6),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_3577_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A),N: A,K2: nat] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),N)) ) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A6)),K2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7121269368397514597t_prod(B,A,F3,A6)),aa(nat,A,power_power(A,N),K2))) ) ) ) ) ).

% prod_le_power
tff(fact_3578_sum__bounded__above__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_field(A)
     => ! [A6: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),K5),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A6))))) )
         => ( pp(aa(set(B),bool,finite_finite2(B),A6))
           => ( ( A6 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups7311177749621191930dd_sum(B,A,F3,A6)),K5)) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3579_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A6: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I3)),K5)) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A6)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),groups7311177749621191930dd_sum(B,A,F3,A6)),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),A6))),K5))) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3580_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_3581_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),A3: B,B2: fun(B,A),C3: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ms(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C3),S) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(nat,A,power_power(A,C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),S)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S))
             => ( groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ms(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C3),S) = aa(nat,A,power_power(A,C3),aa(set(B),nat,finite_card(B),S)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_3582_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F3: fun(A,B),I2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(A,bool)),I5),F3))))
         => ( ( pp(aa(set(A),bool,member(A,I2),I5))
             => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I5)),aa(A,B,F3,I2)) ) )
            & ( ~ pp(aa(set(A),bool,member(A,I2),I5))
             => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F3,I5) ) ) ) ) ) ).

% sum_diff1'
tff(fact_3583_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,B),X5: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),aTP_Lamp_mt(A,option(C))),X5) = none(B) ).

% map_option_o_empty
tff(fact_3584_and__not__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa2) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2)))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = none(num) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one),one))) ) ) )
         => ( ( ( X = one )
             => ! [N4: num] :
                  ( ( Xa2 = aa(num,num,bit0,N4) )
                 => ( ( Y = aa(num,option(num),some(num),one) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one),aa(num,num,bit0,N4)))) ) ) )
           => ( ( ( X = one )
               => ! [N4: num] :
                    ( ( Xa2 = aa(num,num,bit1,N4) )
                   => ( ( Y = none(num) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one),aa(num,num,bit1,N4)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N4: num] :
                          ( ( Xa2 = aa(num,num,bit0,N4) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N4)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N4)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N4: num] :
                            ( ( Xa2 = aa(num,num,bit1,N4) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N4)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N4)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N4: num] :
                                ( ( Xa2 = aa(num,num,bit0,N4) )
                               => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one),aTP_Lamp_es(num,option(num)),bit_and_not_num(M5,N4)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N4)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N4: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N4) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N4)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N4)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_3585_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != aa(num,option(num),some(num),one) ) ) )
       => ( ( ( X = one )
           => ( ? [N4: num] : Xa2 = aa(num,num,bit0,N4)
             => ( Y != none(num) ) ) )
         => ( ( ( X = one )
             => ( ? [N4: num] : Xa2 = aa(num,num,bit1,N4)
               => ( Y != aa(num,option(num),some(num),one) ) ) )
           => ( ( ? [M5: num] : X = aa(num,num,bit0,M5)
               => ( ( Xa2 = one )
                 => ( Y != none(num) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N4: num] :
                        ( ( Xa2 = aa(num,num,bit0,N4) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N4)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N4: num] :
                          ( ( Xa2 = aa(num,num,bit1,N4) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N4)) ) ) )
                 => ( ( ? [M5: num] : X = aa(num,num,bit1,M5)
                     => ( ( Xa2 = one )
                       => ( Y != aa(num,option(num),some(num),one) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N4: num] :
                              ( ( Xa2 = aa(num,num,bit0,N4) )
                             => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N4)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N4: num] :
                                ( ( Xa2 = aa(num,num,bit1,N4) )
                               => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one),aTP_Lamp_es(num,option(num)),bit_un7362597486090784418nd_num(M5,N4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_3586_card__lists__length__le,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_mu(set(A),fun(nat,fun(list(A),bool)),A6),N))) = groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A6)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% card_lists_length_le
tff(fact_3587_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_mv(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A3),Xs),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_3588_Suc__funpow,axiom,
    ! [N: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).

% Suc_funpow
tff(fact_3589_funpow__0,axiom,
    ! [A: $tType,F3: fun(A,A),X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3),X) = X ).

% funpow_0
tff(fact_3590_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B5))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,member(A,X4),B5)) ) ) ).

% subset_code(1)
tff(fact_3591_funpow__swap1,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat,X: A] : aa(A,A,F3,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),aa(A,A,F3,X)) ).

% funpow_swap1
tff(fact_3592_funpow__mult,axiom,
    ! [A: $tType,N: nat,M: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),F3) ).

% funpow_mult
tff(fact_3593_comp__funpow,axiom,
    ! [B: $tType,A: $tType,N: nat,F3: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N),comp(A,A,B,F3)) = comp(A,A,B,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% comp_funpow
tff(fact_3594_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,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,Ys2))))
     => pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),Ys2))) ) ).

% set_zip_rightD
tff(fact_3595_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,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,Ys2))))
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_zip_leftD
tff(fact_3596_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,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,Ys2))))
     => ~ ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),Ys2))) ) ) ).

% in_set_zipE
tff(fact_3597_zip__same,axiom,
    ! [A: $tType,A3: A,B2: A,Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs))))
    <=> ( pp(aa(set(A),bool,member(A,A3),aa(list(A),set(A),set2(A),Xs)))
        & ( A3 = B2 ) ) ) ).

% zip_same
tff(fact_3598_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ~ ! [Y3: B] : ~ pp(aa(set(product_prod(A,B)),bool,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)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_3599_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),Ys2)))
       => ~ ! [X3: A] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_3600_funpow__Suc__right,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)),F3) ).

% funpow_Suc_right
tff(fact_3601_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% funpow.simps(2)
tff(fact_3602_funpow__add,axiom,
    ! [A: $tType,M: nat,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% funpow_add
tff(fact_3603_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) ).

% set_subset_Cons
tff(fact_3604_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys2)
           => ( ( aa(list(A),set(A),set2(A),Ys2) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys2 = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_3605_find__SomeD_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),X) )
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ).

% find_SomeD(2)
tff(fact_3606_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),X),Y))
    <=> ? [F11: fun(nat,A)] :
          ( ( aa(nat,A,F11,zero_zero(nat)) = X )
          & ( aa(nat,A,F11,N) = Y )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,F11,I)),aa(nat,A,F11,aa(nat,nat,suc,I)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_3607_relpowp__bot,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).

% relpowp_bot
tff(fact_3608_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_pos_if_in_set
tff(fact_3609_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Ys2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) )
            & sorted_wrt(A,ord_less_eq(A),Ys2) ) ) ) ).

% sorted_simps(2)
tff(fact_3610_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys2))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Ys2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) )
            & sorted_wrt(A,ord_less(A),Ys2) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_3611_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P2: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),L)))
         => pp(aa(A,bool,P2,X4)) )
    <=> ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,L),I))) ) ) ).

% all_set_conv_nth
tff(fact_3612_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P2,X4)) )
    <=> ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I))) ) ) ).

% all_set_conv_all_nth
tff(fact_3613_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool),X: A] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(A,bool,P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_3614_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_3615_list__ball__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P2: fun(A,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),N))) ) ) ).

% list_ball_nth
tff(fact_3616_nth__mem,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,member(A,aa(nat,A,nth(A,Xs),N)),aa(list(A),set(A),set2(A),Xs))) ) ).

% nth_mem
tff(fact_3617_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_3618_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_3619_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K2: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K2)),A3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A3) ) ).

% numeral_add_unfold_funpow
tff(fact_3620_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K2: num] : aa(num,A,numeral_numeral(A),K2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_3621_finite__lists__length__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_mw(set(A),fun(nat,fun(list(A),bool)),A6),N)))) ) ).

% finite_lists_length_eq
tff(fact_3622_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_3623_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = A6 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != aa(set(A),nat,finite_card(A),A6) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_3624_card__lists__length__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_mw(set(A),fun(nat,fun(list(A),bool)),A6),N))) = aa(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A6)),N) ) ) ).

% card_lists_length_eq
tff(fact_3625_finite__lists__length__le,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_mu(set(A),fun(nat,fun(list(A),bool)),A6),N)))) ) ).

% finite_lists_length_le
tff(fact_3626_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : bit_se5824344872417868541ns_and(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(M,N)) ) ).

% numeral_and_num
tff(fact_3627_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),aa(num,num,bit1,N)) = case_option(option(num),num,aa(num,option(num),some(num),one),aTP_Lamp_es(num,option(num)),bit_un7362597486090784418nd_num(M,N)) ).

% and_num.simps(9)
tff(fact_3628_and__num_Opelims,axiom,
    ! [X: num,Xa2: num,Y: option(num)] :
      ( ( bit_un7362597486090784418nd_num(X,Xa2) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X),Xa2)))
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = aa(num,option(num),some(num),one) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one),one))) ) ) )
         => ( ( ( X = one )
             => ! [N4: num] :
                  ( ( Xa2 = aa(num,num,bit0,N4) )
                 => ( ( Y = none(num) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one),aa(num,num,bit0,N4)))) ) ) )
           => ( ( ( X = one )
               => ! [N4: num] :
                    ( ( Xa2 = aa(num,num,bit1,N4) )
                   => ( ( Y = aa(num,option(num),some(num),one) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one),aa(num,num,bit1,N4)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa2 = one )
                     => ( ( Y = none(num) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),one))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N4: num] :
                          ( ( Xa2 = aa(num,num,bit0,N4) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N4)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit0,N4)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N4: num] :
                            ( ( Xa2 = aa(num,num,bit1,N4) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N4)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M5)),aa(num,num,bit1,N4)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa2 = one )
                           => ( ( Y = aa(num,option(num),some(num),one) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),one))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N4: num] :
                                ( ( Xa2 = aa(num,num,bit0,N4) )
                               => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_un7362597486090784418nd_num(M5,N4)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit0,N4)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N4: num] :
                                  ( ( Xa2 = aa(num,num,bit1,N4) )
                                 => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one),aTP_Lamp_es(num,option(num)),bit_un7362597486090784418nd_num(M5,N4)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M5)),aa(num,num,bit1,N4)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_3629_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),set(A),set2(A),union(A,Xs,Ys2)) = 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),Ys2)) ).

% set_union
tff(fact_3630_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A6: set(A),K2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(set(A),nat,finite_card(A),A6)))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_mx(set(A),fun(nat,fun(list(A),bool)),A6),K2))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_fn(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A6)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A6))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_3631_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K2: nat,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(set(A),nat,finite_card(A),A6)))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),bool),aTP_Lamp_my(nat,fun(set(A),fun(list(A),bool)),K2),A6))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_fn(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A6)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A6))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_3632_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_mx(set(A),fun(nat,fun(list(A),bool)),A6),N)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_3633_distinct__finite__set,axiom,
    ! [A: $tType,X: set(A)] : pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_mz(set(A),fun(list(A),bool),X)))) ).

% distinct_finite_set
tff(fact_3634_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_3635_distinct__length__le,axiom,
    ! [A: $tType,Ys2: list(A),Xs: list(A)] :
      ( distinct(A,Ys2)
     => ( ( aa(list(A),set(A),set2(A),Ys2) = aa(list(A),set(A),set2(A),Xs) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% distinct_length_le
tff(fact_3636_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys2)
             => ( distinct(A,Ys2)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys2) )
                 => ( Xs = Ys2 ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_3637_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
         => ! [J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( I != J )
               => ( aa(nat,A,nth(A,Xs),I) != aa(nat,A,nth(A,Xs),J) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_3638_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,J2: nat] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),J2) )
          <=> ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_3639_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A6 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y5: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y5) = A6 )
                    & sorted_wrt(A,ord_less_eq(A),Y5)
                    & distinct(A,Y5) )
                 => ( Y5 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_3640_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ? [X3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs)))
            & ( aa(nat,A,nth(A,Xs),X3) = X )
            & ! [Y5: nat] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y5),aa(list(A),nat,size_size(list(A)),Xs)))
                  & ( aa(nat,A,nth(A,Xs),Y5) = X ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_3641_distinct__finite__subset,axiom,
    ! [A: $tType,X: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),X))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_na(set(A),fun(list(A),bool),X)))) ) ).

% distinct_finite_subset
tff(fact_3642_distinct__sorted__strict__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I2: nat,J2: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L)))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J2)))
                <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2)) ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_3643_distinct__sorted__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I2: nat,J2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( distinct(A,L)
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J2))) ) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_3644_distinct__sorted__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I2: nat,J2: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),L)))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I2)),aa(nat,A,nth(A,L),J2)))
                <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2)) ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_3645_mergesort__remdups__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( distinct(A,aa(list(A),list(A),mergesort_remdups(A),L))
          & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_remdups(A),L))
          & ( aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_remdups(A),L)) = aa(list(A),set(A),set2(A),L) ) ) ) ).

% mergesort_remdups_correct
tff(fact_3646_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( groups7311177749621191930dd_sum(A,nat,count_list(A,Xs),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_3647_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_nb(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_3648_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_3649_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_nc(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_3650_map__update__eta__repair_I2_J,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K2: A,V2: B] :
      ( ( aa(A,option(B),M,K2) = none(B) )
     => ( ran(A,B,aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_nd(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),M),K2),V2)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),V2),ran(A,B,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_3651_ranI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M,A3) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(A),bool,member(A,B2),ran(B,A,M))) ) ).

% ranI
tff(fact_3652_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).

% count_le_length
tff(fact_3653_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)) ) )
      & ( ( X != Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(A,nat,count_list(A,Xs),Y) ) ) ) ).

% count_list.simps(2)
tff(fact_3654_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( 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),Ys2)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys2)) = aa(nat,nat,binomial(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)),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_3655_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_3656_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),L: list(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A6 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A6) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A6) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_3657_finite__enumerate,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => ? [R: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(nat),nat,finite_card(nat),S)))
          & ! [N8: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N8),aa(set(nat),nat,finite_card(nat),S)))
             => pp(aa(set(nat),bool,member(nat,aa(nat,nat,R,N8)),S)) ) ) ) ).

% finite_enumerate
tff(fact_3658_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or5935395276787703475ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_3659_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K2))
         => ( set_or5935395276787703475ssThan(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_3660_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_3661_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_3662_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioo_iff
tff(fact_3663_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys2: list(A)] :
      ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = 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)),Ys2)) ) ) ).

% length_shuffles
tff(fact_3664_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys2: list(A)] :
      ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
     => ( aa(list(A),set(A),set2(A),Zs) = 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),Ys2)) ) ) ).

% set_shuffles
tff(fact_3665_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B2))) ) ) ).

% infinite_Ioo
tff(fact_3666_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A6)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_3667_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A6)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_3668_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_3669_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_3670_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_3671_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_3672_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_3673_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(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_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_3674_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_3675_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_3676_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_3677_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( strict_mono_on(A,B,F3,A6)
        <=> ! [R4: A,S7: A] :
              ( ( pp(aa(set(A),bool,member(A,R4),A6))
                & pp(aa(set(A),bool,member(A,S7),A6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R4),S7)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R4)),aa(A,B,F3,S7))) ) ) ) ).

% strict_mono_on_def
tff(fact_3678_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A6: set(A),F3: fun(A,B)] :
          ( ! [R: A,S5: A] :
              ( pp(aa(set(A),bool,member(A,R),A6))
             => ( pp(aa(set(A),bool,member(A,S5),A6))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R),S5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R)),aa(A,B,F3,S5))) ) ) )
         => strict_mono_on(A,B,F3,A6) ) ) ).

% strict_mono_onI
tff(fact_3679_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A6: set(A),R3: A,S2: A] :
          ( strict_mono_on(A,B,F3,A6)
         => ( pp(aa(set(A),bool,member(A,R3),A6))
           => ( pp(aa(set(A),bool,member(A,S2),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R3),S2))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R3)),aa(A,B,F3,S2))) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_3680_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F3: fun(A,B),A6: set(A),X: A,Y: A] :
          ( strict_mono_on(A,B,F3,A6)
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( pp(aa(set(A),bool,member(A,Y),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_3681_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or5935395276787703475ssThan(code_integer,L,U))) ).

% finite_greaterThanLessThan_integer
tff(fact_3682_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),J2))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J2)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I2),J2))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_3683_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_3684_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),aa(nat,nat,suc,I2))))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J2))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_3685_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),N)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_3686_option_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: C,Ga: fun(B,C),F3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),comp(option(B),C,option(A),rec_option(C,B,G3,Ga)),map_option(A,B,F3)) = rec_option(C,A,G3,aa(fun(A,B),fun(A,C),aTP_Lamp_ar(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F3)) ).

% option.rec_o_map
tff(fact_3687_split__seed__def,axiom,
    ! [S2: product_prod(code_natural,code_natural)] : split_seed(S2) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_nf(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),S2)),S2) ).

% split_seed_def
tff(fact_3688_complete__linorder__sup__max,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ( sup_sup(A) = ord_max(A) ) ) ).

% complete_linorder_sup_max
tff(fact_3689_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),X: product_prod(A,C),G3: fun(C,B)] :
      ( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),X) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G3),X) )
    <=> ( aa(C,B,F3,aa(product_prod(A,C),C,product_snd(A,C),X)) = aa(C,B,G3,aa(product_prod(A,C),C,product_snd(A,C),X)) ) ) ).

% apsnd_eq_conv
tff(fact_3690_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),X: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F3),X)) = aa(C,A,F3,aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% snd_apsnd
tff(fact_3691_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),F3),product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_3692_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( list_update(A,Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_3693_nth__list__update__eq,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_3694_nth__update__invalid,axiom,
    ! [A: $tType,I2: nat,L: list(A),J2: nat,X: A] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( aa(nat,A,nth(A,list_update(A,L,J2,X)),I2) = aa(nat,A,nth(A,L),I2) ) ) ).

% nth_update_invalid
tff(fact_3695_set__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J2)),J2,aa(nat,A,nth(A,Xs),I2))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_3696_distinct__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( distinct(A,list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J2)),J2,aa(nat,A,nth(A,Xs),I2)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_3697_sndE,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A3: A,B2: B,P2: fun(B,bool)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
     => ( pp(aa(B,bool,P2,aa(product_prod(A,B),B,product_snd(A,B),X)))
       => pp(aa(B,bool,P2,B2)) ) ) ).

% sndE
tff(fact_3698_snd__eqD,axiom,
    ! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
      ( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
tff(fact_3699_snd__conv,axiom,
    ! [Aa: $tType,A: $tType,X1: Aa,X2: A] : aa(product_prod(Aa,A),A,product_snd(Aa,A),aa(A,product_prod(Aa,A),aa(Aa,fun(A,product_prod(Aa,A)),product_Pair(Aa,A),X1),X2)) = X2 ).

% snd_conv
tff(fact_3700_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,X: A,Ys2: list(B),Y: B] : zip(A,B,list_update(A,Xs,I2,X),list_update(B,Ys2,I2,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys2),I2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_update
tff(fact_3701_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ng(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_3702_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I2: nat,V2: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),I2,V2) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_nh(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V2),I2) ).

% list_update.simps(2)
tff(fact_3703_snd__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,aa(fun(A,fun(B,B)),fun(product_prod(A,B),B),product_case_prod(A,B,B),aTP_Lamp_ni(A,fun(B,B))),Prod) ).

% snd_def
tff(fact_3704_fn__snd__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] : aTP_Lamp_nj(fun(B,C),fun(product_prod(A,B),C),F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_nk(fun(B,C),fun(A,fun(B,C)),F3)) ).

% fn_snd_conv
tff(fact_3705_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A6: set(A),X: A,I2: nat] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A6))
     => ( pp(aa(set(A),bool,member(A,X),A6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,X))),A6)) ) ) ).

% set_update_subsetI
tff(fact_3706_option_Osimps_I7_J,axiom,
    ! [C: $tType,A: $tType,F1: C,F22: fun(A,C),X2: A] : aa(option(A),C,rec_option(C,A,F1,F22),aa(A,option(A),some(A),X2)) = aa(A,C,F22,X2) ).

% option.simps(7)
tff(fact_3707_option_Osimps_I6_J,axiom,
    ! [A: $tType,C: $tType,F1: C,F22: fun(A,C)] : aa(option(A),C,rec_option(C,A,F1,F22),none(A)) = F1 ).

% option.simps(6)
tff(fact_3708_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,X))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),Xs)))) ).

% set_update_subset_insert
tff(fact_3709_set__update__memI,axiom,
    ! [A: $tType,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)))) ) ).

% set_update_memI
tff(fact_3710_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I2: nat,L: list(A),X: A,Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y))))
      <=> ( ( X = Y )
          | ! [Y4: A] : pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y4)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3711_in__set__upd__cases,axiom,
    ! [A: $tType,X: A,L: list(A),I2: nat,Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
         => ( X != Y ) )
       => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L))) ) ) ).

% in_set_upd_cases
tff(fact_3712_in__set__upd__eq,axiom,
    ! [A: $tType,I2: nat,L: list(A),X: A,Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y))))
      <=> ( ( X = Y )
          | ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L)))
            & ! [Y4: A] : pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),list_update(A,L,I2,Y4)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3713_nth__list__update,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( ( I2 = J2 )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J2) = X ) )
        & ( ( I2 != J2 )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J2) = aa(nat,A,nth(A,Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_3714_list__update__same__conv,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( list_update(A,Xs,I2,X) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_3715_nth__list__update_H,axiom,
    ! [A: $tType,I2: nat,J2: nat,L: list(A),X: A] :
      ( ( ( ( I2 = J2 )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))) )
       => ( aa(nat,A,nth(A,list_update(A,L,I2,X)),J2) = X ) )
      & ( ~ ( ( I2 = J2 )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))) )
       => ( aa(nat,A,nth(A,list_update(A,L,I2,X)),J2) = aa(nat,A,nth(A,L),J2) ) ) ) ).

% nth_list_update'
tff(fact_3716_insert__swap__set__eq,axiom,
    ! [A: $tType,I2: nat,L: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,L),I2)),aa(list(A),set(A),set2(A),list_update(A,L,I2,X))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_3717_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,Xs: list(A),A3: array(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
         => hoare_hoare_triple(array(A),snga_assn(A,A3,Xs),array_upd(A,I2,X,A3),aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_nl(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I2),Xs),A3),X)) ) ) ).

% upd_rule
tff(fact_3718_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J2))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_3719_subset__mset_Osum__list__update,axiom,
    ! [A: $tType,K2: nat,Xs: list(multiset(A)),X: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),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,K2,X)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),X)),aa(nat,multiset(A),nth(multiset(A),Xs),K2)) ) ) ).

% subset_mset.sum_list_update
tff(fact_3720_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nn(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% times_int.abs_eq
tff(fact_3721_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,member(A,I2),set_or3652927894154168847AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_3722_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K2))
         => ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_3723_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K2,L) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_3724_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_3725_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioc_iff
tff(fact_3726_mod__pure,axiom,
    ! [B2: bool,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(pure_assn(B2)),H))
    <=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) )
        & pp(B2) ) ) ).

% mod_pure
tff(fact_3727_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)) )
    <=> ! [X4: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Ns)))
         => ( X4 = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.sum_list_eq_0_iff
tff(fact_3728_upd__ureturn,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,X: A,A3: array(A)] : heap_Time_bind(array(A),array(A),array_upd(A,I2,X,A3),aTP_Lamp_no(array(A),fun(array(A),heap_Time_Heap(array(A))),A3)) = array_upd(A,I2,X,A3) ) ).

% upd_ureturn
tff(fact_3729_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( set_or3652927894154168847AtMost(A,A3,B2) = set_or3652927894154168847AtMost(A,C3,D3) )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3)) )
            | ( ( A3 = C3 )
              & ( B2 = D3 ) ) ) ) ) ).

% Ioc_inj
tff(fact_3730_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B2))) ) ) ).

% infinite_Ioc
tff(fact_3731_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_3732_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_3733_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)) = bot_bot(set(A)) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),A3)) ) ) ) ).

% Ioc_disjoint
tff(fact_3734_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_3735_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_3736_mod__emp,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(one_one(assn)),H))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) ) ) ).

% mod_emp
tff(fact_3737_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7311177749621191930dd_sum(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),groups7311177749621191930dd_sum(nat,A,G3,set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% sum.head
tff(fact_3738_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( groups7121269368397514597t_prod(nat,A,G3,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),groups7121269368397514597t_prod(nat,A,G3,set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% prod.head
tff(fact_3739_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_3740_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_3741_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_3742_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_3743_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_3744_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_3745_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,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_np(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_3746_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I2)),J2))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J2)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I2),J2))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_3747_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_3748_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_3749_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_3750_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: product_prod(nat,nat)] : aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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_nq(nat,fun(nat,A))),X) ) ).

% of_int.abs_eq
tff(fact_3751_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ns(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X)) ) ).

% less_int.abs_eq
tff(fact_3752_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_nu(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_3753_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_3754_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ny(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_3755_upd_H__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),I2: code_integer,X: A] : array_upd2(A,A3,I2,X) = heap_Time_bind(array(A),product_unit,array_upd(A,code_nat_of_integer(I2),X,A3),aTP_Lamp_nz(array(A),heap_Time_Heap(product_unit))) ) ).

% upd'_def
tff(fact_3756_subset__mset_Oelem__le__sum__list,axiom,
    ! [A: $tType,K2: nat,Ns: list(multiset(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Ns)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),nth(multiset(A),Ns),K2)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns))) ) ).

% subset_mset.elem_le_sum_list
tff(fact_3757_rat__minus__code,axiom,
    ! [P3: rat,Q5: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P3),Q5)) = 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_ob(rat,fun(int,fun(int,product_prod(int,int))),Q5)),quotient_of(P3)) ).

% rat_minus_code
tff(fact_3758_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,M: nat,A3: A] : bit_se4730199178511100633sh_bit(A,N,aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4197421643247451524op_bit(A,N,A3))) = bit_se5824344872417868541ns_and(A,A3,bit_se5824344872417868541ns_and(A,bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),bit_ri4277139882892585799ns_not(A,bit_se2239418461657761734s_mask(A,N)))) ) ).

% slice_eq_mask
tff(fact_3759_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : pp(aa(set(code_integer),bool,finite_finite2(code_integer),set_or3652927894154168847AtMost(code_integer,L,U))) ).

% finite_greaterThanAtMost_integer
tff(fact_3760_mset__subset__eq__mono__add__right__cancel,axiom,
    ! [A: $tType,A6: multiset(A),C4: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5)) ) ).

% mset_subset_eq_mono_add_right_cancel
tff(fact_3761_mset__subset__eq__mono__add__left__cancel,axiom,
    ! [A: $tType,C4: multiset(A),A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),A6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),B5)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5)) ) ).

% mset_subset_eq_mono_add_left_cancel
tff(fact_3762_subset__mset_Oadd__le__cancel__right,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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)),B2),C3)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% subset_mset.add_le_cancel_right
tff(fact_3763_subset__mset_Oadd__le__cancel__left,axiom,
    ! [A: $tType,C3: multiset(A),A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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),B2)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% subset_mset.add_le_cancel_left
tff(fact_3764_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se4197421643247451524op_bit(int,N,K2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_3765_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se4197421643247451524op_bit(int,N,K2)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int))) ) ).

% drop_bit_negative_int_iff
tff(fact_3766_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : bit_se4197421643247451524op_bit(A,M,bit_se4197421643247451524op_bit(A,N,A3)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),A3) ) ).

% drop_bit_drop_bit
tff(fact_3767_subset__mset_Ole__add__same__cancel2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2)) ) ).

% subset_mset.le_add_same_cancel2
tff(fact_3768_subset__mset_Ole__add__same__cancel1,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2)) ) ).

% subset_mset.le_add_same_cancel1
tff(fact_3769_subset__mset_Oadd__le__same__cancel2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),B2))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A)))) ) ).

% subset_mset.add_le_same_cancel2
tff(fact_3770_subset__mset_Oadd__le__same__cancel1,axiom,
    ! [A: $tType,B2: multiset(A),A3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3)),B2))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A)))) ) ).

% subset_mset.add_le_same_cancel1
tff(fact_3771_subset__mset_Odiff__add,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( 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)),minus_minus(multiset(A)),B2),A3)),A3) = B2 ) ) ).

% subset_mset.diff_add
tff(fact_3772_subset__mset_Oadd__diff__assoc,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),C3),B2)),A3) ) ) ).

% subset_mset.add_diff_assoc
tff(fact_3773_subset__mset_Oadd__diff__assoc2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( 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)),minus_minus(multiset(A)),B2),A3)),C3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(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),C3)),A3) ) ) ).

% subset_mset.add_diff_assoc2
tff(fact_3774_mset__subset__eq__multiset__union__diff__commute,axiom,
    ! [A: $tType,B5: multiset(A),A6: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),A6))
     => ( 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)),minus_minus(multiset(A)),A6),B5)),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),A6),C4)),B5) ) ) ).

% mset_subset_eq_multiset_union_diff_commute
tff(fact_3775_subset__mset_Oadd__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A),D3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C3),D3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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)),B2),D3))) ) ) ).

% subset_mset.add_mono
tff(fact_3776_subset__mset_Oless__eqE,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ~ ! [C2: multiset(A)] : B2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2) ) ).

% subset_mset.less_eqE
tff(fact_3777_subset__mset_Ole__iff__add,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
    <=> ? [C5: multiset(A)] : B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C5) ) ).

% subset_mset.le_iff_add
tff(fact_3778_subset__mset_Oadd__left__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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),B2))) ) ).

% subset_mset.add_left_mono
tff(fact_3779_subset__mset_Oadd__right__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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)),B2),C3))) ) ).

% subset_mset.add_right_mono
tff(fact_3780_subset__mset_Oadd__le__imp__le__left,axiom,
    ! [A: $tType,C3: multiset(A),A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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),B2)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% subset_mset.add_le_imp_le_left
tff(fact_3781_subset__mset_Oadd__le__imp__le__right,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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)),B2),C3)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% subset_mset.add_le_imp_le_right
tff(fact_3782_mset__subset__eq__add__left,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5))) ).

% mset_subset_eq_add_left
tff(fact_3783_mset__subset__eq__mono__add,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),C4: multiset(A),D5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C4),D5))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),D5))) ) ) ).

% mset_subset_eq_mono_add
tff(fact_3784_mset__subset__eq__add__right,axiom,
    ! [A: $tType,B5: multiset(A),A6: multiset(A)] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5))) ).

% mset_subset_eq_add_right
tff(fact_3785_mset__subset__eq__exists__conv,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5))
    <=> ? [C6: multiset(A)] : B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),C6) ) ).

% mset_subset_eq_exists_conv
tff(fact_3786_mset__le__incr__right2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),B2))) ) ).

% mset_le_incr_right2
tff(fact_3787_mset__le__incr__right1,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C3))) ) ).

% mset_le_incr_right1
tff(fact_3788_mset__le__decr__left2,axiom,
    ! [A: $tType,C3: multiset(A),A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),A3)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% mset_le_decr_left2
tff(fact_3789_mset__le__decr__left1,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% mset_le_decr_left1
tff(fact_3790_mset__union__subset,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)),C4))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),C4))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),C4)) ) ) ).

% mset_union_subset
tff(fact_3791_mset__le__distrib,axiom,
    ! [A: $tType,X6: multiset(A),A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)))
     => ~ ! [Xa5: multiset(A),Xb4: multiset(A)] :
            ( ( X6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa5),Xb4) )
           => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xa5),A6))
             => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xb4),B5)) ) ) ) ).

% mset_le_distrib
tff(fact_3792_mset__le__addE,axiom,
    ! [A: $tType,Xs: multiset(A),Ys2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xs),Ys2))
     => ~ ! [Zs2: multiset(A)] : Ys2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xs),Zs2) ) ).

% mset_le_addE
tff(fact_3793_diff__rat__def,axiom,
    ! [Q5: rat,R3: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q5),R3) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q5),aa(rat,rat,uminus_uminus(rat),R3)) ).

% diff_rat_def
tff(fact_3794_mset__le__subtract,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A6),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C4))) ) ).

% mset_le_subtract
tff(fact_3795_subset__mset_Olift__Suc__antimono__le,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N: nat,N2: nat] :
      ( ! [N4: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,aa(nat,nat,suc,N4))),aa(nat,multiset(A),F3,N4)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,N2)),aa(nat,multiset(A),F3,N))) ) ) ).

% subset_mset.lift_Suc_antimono_le
tff(fact_3796_subset__mset_Olift__Suc__mono__le,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N: nat,N2: nat] :
      ( ! [N4: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,N4)),aa(nat,multiset(A),F3,aa(nat,nat,suc,N4))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),F3,N)),aa(nat,multiset(A),F3,N2))) ) ) ).

% subset_mset.lift_Suc_mono_le
tff(fact_3797_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),zero_zero(multiset(A))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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)),X),Y) = zero_zero(multiset(A)) )
        <=> ( ( X = zero_zero(multiset(A)) )
            & ( Y = zero_zero(multiset(A)) ) ) ) ) ) ).

% subset_mset.add_nonpos_eq_0_iff
tff(fact_3798_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),X))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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)),X),Y) = zero_zero(multiset(A)) )
        <=> ( ( X = zero_zero(multiset(A)) )
            & ( Y = zero_zero(multiset(A)) ) ) ) ) ) ).

% subset_mset.add_nonneg_eq_0_iff
tff(fact_3799_subset__mset_Oadd__nonpos__nonpos,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),zero_zero(multiset(A))))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),zero_zero(multiset(A)))) ) ) ).

% subset_mset.add_nonpos_nonpos
tff(fact_3800_subset__mset_Oadd__nonneg__nonneg,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2))) ) ) ).

% subset_mset.add_nonneg_nonneg
tff(fact_3801_subset__mset_Oadd__increasing2,axiom,
    ! [A: $tType,C3: multiset(A),B2: multiset(A),A3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),C3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),A3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3))) ) ) ).

% subset_mset.add_increasing2
tff(fact_3802_subset__mset_Oadd__decreasing2,axiom,
    ! [A: $tType,C3: multiset(A),A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C3),zero_zero(multiset(A))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3)),B2)) ) ) ).

% subset_mset.add_decreasing2
tff(fact_3803_subset__mset_Oadd__increasing,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),C3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3))) ) ) ).

% subset_mset.add_increasing
tff(fact_3804_subset__mset_Oadd__decreasing,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C3),B2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3)),B2)) ) ) ).

% subset_mset.add_decreasing
tff(fact_3805_subset__mset_Ole__add__diff,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C3),aa(multiset(A),multiset(A),aa(multiset(A),fun(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),C3)),A3))) ) ).

% subset_mset.le_add_diff
tff(fact_3806_subset__mset_Ole__diff__conv2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),A3)),B2)) ) ) ).

% subset_mset.le_diff_conv2
tff(fact_3807_subset__mset_Odiff__add__assoc,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),C3),B2)),A3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) ) ) ).

% subset_mset.diff_add_assoc
tff(fact_3808_subset__mset_Odiff__add__assoc2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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),C3)),A3) = 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)),minus_minus(multiset(A)),B2),A3)),C3) ) ) ).

% subset_mset.diff_add_assoc2
tff(fact_3809_subset__mset_Odiff__diff__right,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),C3),A3)),B2) ) ) ).

% subset_mset.diff_diff_right
tff(fact_3810_subset__mset_Oadd__diff__inverse,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) = B2 ) ) ).

% subset_mset.add_diff_inverse
tff(fact_3811_subset__mset_Ole__imp__diff__is__add,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
       => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3) = C3 )
        <=> ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),A3) ) ) ) ) ).

% subset_mset.le_imp_diff_is_add
tff(fact_3812_subset__eq__diff__conv,axiom,
    ! [A: $tType,A6: multiset(A),C4: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A6),C4)),B5))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4))) ) ).

% subset_eq_diff_conv
tff(fact_3813_multiset__diff__union__assoc,axiom,
    ! [A: $tType,C4: multiset(A),B5: multiset(A),A6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C4),B5))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),A6),B5)),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C4)) ) ) ).

% multiset_diff_union_assoc
tff(fact_3814_mset__le__subtract__right,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),B5)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),X6)) ) ) ).

% mset_le_subtract_right
tff(fact_3815_mset__le__subtract__left,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),A6)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),X6)) ) ) ).

% mset_le_subtract_left
tff(fact_3816_size__mset__mono,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),A6)),aa(multiset(A),nat,size_size(multiset(A)),B5))) ) ).

% size_mset_mono
tff(fact_3817_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se4197421643247451524op_bit(A,N,A3)) = bit_se4197421643247451524op_bit(A,N,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3)) ) ).

% take_bit_drop_bit
tff(fact_3818_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5641148757651400278ts_bit(A,bit_se4197421643247451524op_bit(A,N,A3)) = aa(fun(nat,nat),fun(nat,bool),comp(nat,bool,nat,bit_se5641148757651400278ts_bit(A,A3)),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_drop_bit_eq
tff(fact_3819_quotient__of__denom__pos_H,axiom,
    ! [R3: rat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R3)))) ).

% quotient_of_denom_pos'
tff(fact_3820_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_3821_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,N,bit_se4197421643247451524op_bit(A,N,A3))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = A3 ) ).

% bits_ident
tff(fact_3822_quotient__of__denom__pos,axiom,
    ! [R3: rat,P3: int,Q5: int] :
      ( ( quotient_of(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q5) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q5)) ) ).

% quotient_of_denom_pos
tff(fact_3823_subset__mset_Osum__list__nonneg,axiom,
    ! [A: $tType,Xs: list(multiset(A))] :
      ( ! [X3: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),X3)) )
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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_3824_subset__mset_Osum__list__nonpos,axiom,
    ! [A: $tType,Xs: list(multiset(A))] :
      ( ! [X3: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X3),zero_zero(multiset(A)))) )
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),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_3825_subset__mset_Omember__le__sum__list,axiom,
    ! [A: $tType,X: multiset(A),Xs: list(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,member(multiset(A),X),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs))) ) ).

% subset_mset.member_le_sum_list
tff(fact_3826_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType,Xs: list(multiset(A))] :
      ( ! [X3: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X3),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),X3)) )
     => ( ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs) = zero_zero(multiset(A)) )
      <=> ! [X4: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
           => ( X4 = zero_zero(multiset(A)) ) ) ) ) ).

% subset_mset.sum_list_nonneg_eq_0_iff
tff(fact_3827_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_3828_rat__uminus__code,axiom,
    ! [P3: rat] : quotient_of(aa(rat,rat,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_oc(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_uminus_code
tff(fact_3829_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_od(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_abs_code
tff(fact_3830_rat__less__eq__code,axiom,
    ! [P3: rat,Q5: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),P3),Q5))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_of(rat,fun(int,fun(int,bool)),Q5)),quotient_of(P3))) ) ).

% rat_less_eq_code
tff(fact_3831_rat__less__code,axiom,
    ! [P3: rat,Q5: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P3),Q5))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_oh(rat,fun(int,fun(int,bool)),Q5)),quotient_of(P3))) ) ).

% rat_less_code
tff(fact_3832_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_3833_rat__divide__code,axiom,
    ! [P3: rat,Q5: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q5)) = 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_oj(rat,fun(int,fun(int,product_prod(int,int))),Q5)),quotient_of(P3)) ).

% rat_divide_code
tff(fact_3834_rat__times__code,axiom,
    ! [P3: rat,Q5: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q5)) = 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_ol(rat,fun(int,fun(int,product_prod(int,int))),Q5)),quotient_of(P3)) ).

% rat_times_code
tff(fact_3835_rat__plus__code,axiom,
    ! [P3: rat,Q5: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q5)) = 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_on(rat,fun(int,fun(int,product_prod(int,int))),Q5)),quotient_of(P3)) ).

% rat_plus_code
tff(fact_3836_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_oo(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).

% rat_inverse_code
tff(fact_3837_subset__mset_Osum__mono2,axiom,
    ! [A: $tType,B: $tType,B5: set(B),A6: set(B),F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),B5))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
       => ( ! [B4: B] :
              ( pp(aa(set(B),bool,member(B,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A6)))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,B4))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,A6)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,B5))) ) ) ) ).

% subset_mset.sum_mono2
tff(fact_3838_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_nu(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_3839_less__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),Xa2))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ns(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).

% less_int.rep_eq
tff(fact_3840_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3841_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3842_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% inverse_positive_iff_positive
tff(fact_3843_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_3844_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3845_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3846_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3847_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3848_subset__mset_Osum__eq__0__iff,axiom,
    ! [B: $tType,A: $tType,F4: set(B),F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),F4))
     => ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,F4) = zero_zero(multiset(A)) )
      <=> ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),F4))
           => ( aa(B,multiset(A),F3,X4) = zero_zero(multiset(A)) ) ) ) ) ).

% subset_mset.sum_eq_0_iff
tff(fact_3849_abs__rat__def,axiom,
    ! [A3: rat] :
      ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A3) = aa(rat,rat,uminus_uminus(rat),A3) ) )
      & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A3) = A3 ) ) ) ).

% abs_rat_def
tff(fact_3850_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),X),Y))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_3851_sgn__rat__def,axiom,
    ! [A3: rat] :
      ( ( ( A3 = zero_zero(rat) )
       => ( aa(rat,rat,sgn_sgn(rat),A3) = zero_zero(rat) ) )
      & ( ( A3 != zero_zero(rat) )
       => ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
           => ( aa(rat,rat,sgn_sgn(rat),A3) = one_one(rat) ) )
          & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
           => ( aa(rat,rat,sgn_sgn(rat),A3) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_3852_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P2: fun(B,bool),Q2: fun(B,bool),G3: fun(A,B),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,aa(fun(B,bool),fun(B,bool),aTP_Lamp_op(fun(B,bool),fun(fun(B,bool),fun(B,bool)),P2),Q2)),G3),X5))
    <=> ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,P2),G3),X5))
        & pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,Q2),G3),X5)) ) ) ).

% conj_comp_iff
tff(fact_3853_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ).

% positive_imp_inverse_positive
tff(fact_3854_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_3855_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3856_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3857_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3858_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3859_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less
tff(fact_3860_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less
tff(fact_3861_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le
tff(fact_3862_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% le_imp_inverse_le
tff(fact_3863_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3864_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_3865_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% inverse_le_1_iff
tff(fact_3866_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% one_less_inverse_iff
tff(fact_3867_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_less_inverse
tff(fact_3868_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),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),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_3869_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),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),A3),B2)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_3870_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_3871_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_3872_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_le_inverse
tff(fact_3873_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% inverse_less_1_iff
tff(fact_3874_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).

% one_le_inverse_iff
tff(fact_3875_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N4)))),X)) ) ) ).

% reals_Archimedean
tff(fact_3876_subset__mset_Osum__nonneg,axiom,
    ! [A: $tType,B: $tType,A6: set(B),F3: fun(B,multiset(A))] :
      ( ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A6))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,X3))) )
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,A6))) ) ).

% subset_mset.sum_nonneg
tff(fact_3877_subset__mset_Osum__nonpos,axiom,
    ! [B: $tType,A: $tType,A6: set(B),F3: fun(B,multiset(A))] :
      ( ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A6))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,X3)),zero_zero(multiset(A)))) )
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,A6)),zero_zero(multiset(A)))) ) ).

% subset_mset.sum_nonpos
tff(fact_3878_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N4))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3879_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(nat,A,power_power(A,X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),N)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3880_subset__mset_Osum__mono,axiom,
    ! [A: $tType,B: $tType,K5: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( ! [I3: B] :
          ( pp(aa(set(B),bool,member(B,I3),K5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,I3)),aa(B,multiset(A),G3,I3))) )
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,K5)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),G3,K5))) ) ).

% subset_mset.sum_mono
tff(fact_3881_subset__mset_Osum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType,A6: set(B),F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),A6))
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A6))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,X3))) )
       => ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,A6) = zero_zero(multiset(A)) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
             => ( aa(B,multiset(A),F3,X4) = zero_zero(multiset(A)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_eq_0_iff
tff(fact_3882_subset__mset_Osum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(B),T5: set(C),G3: fun(C,multiset(A)),I2: fun(C,B),F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),S2))
     => ( pp(aa(set(C),bool,finite_finite2(C),T5))
       => ( ! [X3: C] :
              ( pp(aa(set(C),bool,member(C,X3),T5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(C,multiset(A),G3,X3))) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),S2))
               => ? [Xa: C] :
                    ( pp(aa(set(C),bool,member(C,Xa),T5))
                    & ( aa(C,B,I2,Xa) = X3 )
                    & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,X3)),aa(C,multiset(A),G3,Xa))) ) )
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,S2)),groups3894954378712506084id_sum(multiset(A),C,plus_plus(multiset(A)),zero_zero(multiset(A)),G3,T5))) ) ) ) ) ).

% subset_mset.sum_le_included
tff(fact_3883_subset__mset_Osum__nonneg__0,axiom,
    ! [B: $tType,A: $tType,S2: set(B),F3: fun(B,multiset(A)),I2: B] :
      ( pp(aa(set(B),bool,finite_finite2(B),S2))
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),S2))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I3))) )
       => ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,S2) = zero_zero(multiset(A)) )
         => ( pp(aa(set(B),bool,member(B,I2),S2))
           => ( aa(B,multiset(A),F3,I2) = zero_zero(multiset(A)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_0
tff(fact_3884_subset__mset_Osum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType,S2: set(B),F3: fun(B,multiset(A)),B5: multiset(A),I2: B] :
      ( pp(aa(set(B),bool,finite_finite2(B),S2))
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),S2))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I3))) )
       => ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,S2) = B5 )
         => ( pp(aa(set(B),bool,member(B,I2),S2))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,I2)),B5)) ) ) ) ) ).

% subset_mset.sum_nonneg_leq_bound
tff(fact_3885_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),X) = 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_nq(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).

% of_int.rep_eq
tff(fact_3886_mset__size2elem,axiom,
    ! [A: $tType,P2: multiset(A),Q5: A,Q6: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q5),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q6),zero_zero(multiset(A))))),P2))
       => ( P2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q5),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q6),zero_zero(multiset(A)))) ) ) ) ).

% mset_size2elem
tff(fact_3887_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_oq(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_3888_uminus__int__def,axiom,
    uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),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_np(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_3889_in__set__enumerate__eq,axiom,
    ! [A: $tType,P3: product_prod(nat,A),N: nat,Xs: list(A)] :
      ( pp(aa(set(product_prod(nat,A)),bool,member(product_prod(nat,A),P3),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).

% in_set_enumerate_eq
tff(fact_3890_union__mset__add__mset__right,axiom,
    ! [A: $tType,A6: multiset(A),A3: A,B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),B5)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)) ).

% union_mset_add_mset_right
tff(fact_3891_union__mset__add__mset__left,axiom,
    ! [A: $tType,A3: A,A6: multiset(A),B5: 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(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A6)),B5) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)) ).

% union_mset_add_mset_left
tff(fact_3892_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),X: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),X)) = aa(product_prod(A,C),A,product_fst(A,C),X) ).

% fst_apsnd
tff(fact_3893_prod_Ocollapse,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).

% prod.collapse
tff(fact_3894_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_3895_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S2: product_prod(A,B),T5: product_prod(A,B)] :
      ( ( S2 = T5 )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S2) = aa(product_prod(A,B),A,product_fst(A,B),T5) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S2) = aa(product_prod(A,B),B,product_snd(A,B),T5) ) ) ) ).

% prod_eq_iff
tff(fact_3896_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] :
      ( ! [A9: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P2,A9),X_12))
    <=> ! [Z2: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(A,B),A,product_fst(A,B),Z2)),aa(product_prod(A,B),B,product_snd(A,B),Z2))) ) ).

% All_prod_contract
tff(fact_3897_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),Q5: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P3) = aa(product_prod(A,B),A,product_fst(A,B),Q5) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P3) = aa(product_prod(A,B),B,product_snd(A,B),Q5) )
       => ( P3 = Q5 ) ) ) ).

% prod_eqI
tff(fact_3898_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] :
      ( ? [A9: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P2,A9),X_12))
    <=> ? [Z2: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(A,B),A,product_fst(A,B),Z2)),aa(product_prod(A,B),B,product_snd(A,B),Z2))) ) ).

% Ex_prod_contract
tff(fact_3899_prod_Oexpand,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
      ( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_3900_mset__union__2__elem,axiom,
    ! [A: $tType,A3: A,B2: A,C3: A,M6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C3),M6) )
     => ( ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = M6 )
          & ( B2 = C3 ) )
        | ( ( A3 = C3 )
          & ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))) = M6 ) ) ) ) ).

% mset_union_2_elem
tff(fact_3901_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X2: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2)) = X1 ).

% fst_conv
tff(fact_3902_fst__eqD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
tff(fact_3903_fstE,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),A3: A,B2: B,P2: fun(A,bool)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
     => ( pp(aa(A,bool,P2,aa(product_prod(A,B),A,product_fst(A,B),X)))
       => pp(aa(A,bool,P2,A3)) ) ) ).

% fstE
tff(fact_3904_mset__le__add__mset__decr__left1,axiom,
    ! [A: $tType,C3: A,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C3),A3)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).

% mset_le_add_mset_decr_left1
tff(fact_3905_fst__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_or(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_3906_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] : aTP_Lamp_os(fun(A,C),fun(product_prod(A,B),C),F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_ot(fun(A,C),fun(A,fun(B,C)),F3)) ).

% fn_fst_conv
tff(fact_3907_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),Y: A,X: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Y),X))
     => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)))) ) ).

% exI_realizer
tff(fact_3908_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),P3: A,Q2: fun(B,bool),Q5: B] :
      ( pp(aa(A,bool,P2,P3))
     => ( pp(aa(B,bool,Q2,Q5))
       => ( pp(aa(A,bool,P2,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q5))))
          & pp(aa(B,bool,Q2,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q5)))) ) ) ) ).

% conjI_realizer
tff(fact_3909_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),X: A,Y: B,A3: product_prod(A,B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X),Y))
     => ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
       => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3))) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_3910_prod_Oexhaust__sel,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% prod.exhaust_sel
tff(fact_3911_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T5: product_prod(A,B)] : T5 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T5)),aa(product_prod(A,B),B,product_snd(A,B),T5)) ).

% surjective_pairing
tff(fact_3912_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),P3: product_prod(B,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),F3),P3) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),P3)),aa(product_prod(B,C),C,product_snd(B,C),P3)) ).

% case_prod_beta
tff(fact_3913_split__beta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% split_beta
tff(fact_3914_mset__le__add__mset__decr__left2,axiom,
    ! [A: $tType,C3: A,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C3),A3)),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C3),zero_zero(multiset(A)))),B2)) ) ).

% mset_le_add_mset_decr_left2
tff(fact_3915_mset__le__single__cases,axiom,
    ! [A: $tType,M6: multiset(A),A3: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))))
     => ( ( M6 != zero_zero(multiset(A)) )
       => ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ) ).

% mset_le_single_cases
tff(fact_3916_mset__le__add__mset,axiom,
    ! [A: $tType,X: A,B5: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),B5)),C4))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))),C4))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),C4)) ) ) ).

% mset_le_add_mset
tff(fact_3917_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),A6: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))))
     => pp(aa(B,bool,aa(A,fun(B,bool),A6,aa(product_prod(A,B),A,product_fst(A,B),X)),aa(product_prod(A,B),B,product_snd(A,B),X))) ) ).

% Product_Type.Collect_case_prodD
tff(fact_3918_add__mset__replicate__mset__safe,axiom,
    ! [A: $tType,B: $tType,M6: multiset(B),A3: B] :
      ( nO_MATCH(multiset(A),multiset(B),zero_zero(multiset(A)),M6)
     => ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),M6) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),zero_zero(multiset(B)))),M6) ) ) ).

% add_mset_replicate_mset_safe
tff(fact_3919_add__mset__add__single,axiom,
    ! [A: $tType,A3: A,A6: multiset(A)] : aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))) ).

% add_mset_add_single
tff(fact_3920_union__is__single,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),A3: 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(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) )
    <=> ( ( ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) )
          & ( N7 = zero_zero(multiset(A)) ) )
        | ( ( M6 = zero_zero(multiset(A)) )
          & ( N7 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ) ) ).

% union_is_single
tff(fact_3921_single__is__union,axiom,
    ! [A: $tType,A3: A,M6: multiset(A),N7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(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(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = M6 )
          & ( N7 = zero_zero(multiset(A)) ) )
        | ( ( M6 = zero_zero(multiset(A)) )
          & ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = N7 ) ) ) ) ).

% single_is_union
tff(fact_3922_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P2: fun(A,fun(B,bool)),P3: product_prod(B,A),Q2: fun(C,bool),F3: fun(B,fun(A,C))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(B,A),A,product_snd(B,A),P3)),aa(product_prod(B,A),B,product_fst(B,A),P3)))
     => ( ! [X3: B,Y3: A] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Y3),X3))
           => pp(aa(C,bool,Q2,aa(A,C,aa(B,fun(A,C),F3,X3),Y3))) )
       => pp(aa(C,bool,Q2,aa(product_prod(B,A),C,aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),F3),P3))) ) ) ).

% exE_realizer
tff(fact_3923_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,F3: fun(A,fun(B,C)),G3: fun(D,A)] : aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_ou(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),F3),G3) = aa(fun(D,fun(B,C)),fun(product_prod(D,B),C),product_case_prod(D,B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_ov(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),F3),G3)) ).

% split_comp_eq
tff(fact_3924_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),X5: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),X5) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),X5)),aa(product_prod(A,B),B,product_snd(A,B),X5)) ).

% case_prod_beta'
tff(fact_3925_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X5: fun(A,fun(B,C)),Xa: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X5),Xa) = aa(B,C,aa(A,fun(B,C),X5,aa(product_prod(A,B),A,product_fst(A,B),Xa)),aa(product_prod(A,B),B,product_snd(A,B),Xa)) ).

% case_prod_unfold
tff(fact_3926_Id__fstsnd__eq,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_ow(product_prod(A,A),bool)) ).

% Id_fstsnd_eq
tff(fact_3927_prod_Osplit__sel,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(C,bool),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P2,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)))
    <=> ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
       => pp(aa(C,bool,P2,aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel
tff(fact_3928_prod_Osplit__sel__asm,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(C,bool),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P2,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)))
    <=> ~ ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
          & ~ pp(aa(C,bool,P2,aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel_asm
tff(fact_3929_scomp__unfold,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,X5: fun(A,product_prod(B,C)),Xa: fun(B,fun(C,D)),Xb: A] : aa(A,D,product_scomp(A,B,C,D,X5,Xa),Xb) = aa(C,D,aa(B,fun(C,D),Xa,aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),X5,Xb))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X5,Xb))) ).

% scomp_unfold
tff(fact_3930_mset__le__subtract__add__mset__left,axiom,
    ! [A: $tType,X: A,B5: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),B5)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A))))))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))),X6)) ) ) ).

% mset_le_subtract_add_mset_left
tff(fact_3931_mset__le__subtract__add__mset__right,axiom,
    ! [A: $tType,X: A,B5: multiset(A),X6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),B5)),X6))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),B5)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),X6)) ) ) ).

% mset_le_subtract_add_mset_right
tff(fact_3932_mset__unplusm__dist__cases2,axiom,
    ! [A: $tType,B5: multiset(A),C4: multiset(A),S2: A,A6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),A6) )
     => ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
         => ( A6 != 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)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))),C4) ) )
       => ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
           => ( A6 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases2
tff(fact_3933_mset__unplusm__dist__cases,axiom,
    ! [A: $tType,S2: A,A6: multiset(A),B5: multiset(A),C4: 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(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),A6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4) )
     => ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
         => ( A6 != 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)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))),C4) ) )
       => ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
           => ( A6 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases
tff(fact_3934_mset__single__cases2_H,axiom,
    ! [A: $tType,S2: A,C3: multiset(A),R6: A,C7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C3) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C7) )
     => ( ( ( S2 = R6 )
         => ( C3 != C7 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) )
             => ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) = Cc )
                 => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases2'
tff(fact_3935_mset__single__cases2,axiom,
    ! [A: $tType,S2: A,C3: multiset(A),R6: A,C7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C3) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C7) )
     => ( ( ( S2 = R6 )
         => ( C3 != C7 ) )
       => ~ ( ( C7 = 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)),minus_minus(multiset(A)),C7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) )
           => ( ( C3 = 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)),minus_minus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases2
tff(fact_3936_mset__single__cases_H,axiom,
    ! [A: $tType,S2: A,C3: multiset(A),R6: A,C7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C3) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C7) )
     => ( ( ( S2 = R6 )
         => ( C3 != C7 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),Cc) )
             => ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))),Cc) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) = Cc )
                 => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases'
tff(fact_3937_mset__single__cases,axiom,
    ! [A: $tType,S2: A,C3: multiset(A),R6: A,C7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),C3) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),C7) )
     => ( ( ( S2 = R6 )
         => ( C3 != C7 ) )
       => ~ ( ( C7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A))))) )
           => ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A))))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R6),zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S2),zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases
tff(fact_3938_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(D,fun(C,A)),G3: fun(B,D),X: product_prod(B,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),aa(fun(B,D),fun(B,fun(C,A)),comp(D,fun(C,A),B,F3),G3)),X) = aa(C,A,aa(D,fun(C,A),F3,aa(B,D,G3,aa(product_prod(B,C),B,product_fst(B,C),X))),aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% case_prod_comp
tff(fact_3939_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ox(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_3940_size__mset__SucE,axiom,
    ! [A: $tType,A6: multiset(A),N: nat] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),A6) = aa(nat,nat,suc,N) )
     => ~ ! [A5: A,B8: multiset(A)] :
            ( ( A6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A5),zero_zero(multiset(A)))),B8) )
           => ( aa(multiset(A),nat,size_size(multiset(A)),B8) != N ) ) ) ).

% size_mset_SucE
tff(fact_3941_size__Diff1__le,axiom,
    ! [A: $tType,M6: multiset(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ).

% size_Diff1_le
tff(fact_3942_the__elem__def,axiom,
    ! [A: $tType,X6: set(A)] : the_elem(A,X6) = the(A,aTP_Lamp_oy(set(A),fun(A,bool),X6)) ).

% the_elem_def
tff(fact_3943_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ng(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_3944_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ox(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_3945_mset__size__le1__cases,axiom,
    ! [A: $tType,M6: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),M6)),aa(nat,nat,suc,zero_zero(nat))))
     => ( ( M6 != zero_zero(multiset(A)) )
       => ~ ! [M5: A] : M6 != aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M5),zero_zero(multiset(A))) ) ) ).

% mset_size_le1_cases
tff(fact_3946_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),P3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
    <=> ? [N3: nat] :
          ( ( aa(nat,A,nth(A,Xs),N3) = aa(product_prod(A,B),A,product_fst(A,B),P3) )
          & ( aa(nat,B,nth(B,Ys2),N3) = aa(product_prod(A,B),B,product_snd(A,B),P3) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(B),nat,size_size(list(B)),Ys2))) ) ) ).

% in_set_zip
tff(fact_3947_old_Orec__unit__def,axiom,
    ! [T: $tType,X5: T,Xa: product_unit] : product_rec_unit(T,X5,Xa) = the(T,product_rec_set_unit(T,X5,Xa)) ).

% old.rec_unit_def
tff(fact_3948_old_Orec__prod__def,axiom,
    ! [T: $tType,B: $tType,A: $tType,X5: fun(A,fun(B,T)),Xa: product_prod(A,B)] : product_rec_prod(A,B,T,X5,Xa) = the(T,product_rec_set_prod(A,B,T,X5,Xa)) ).

% old.rec_prod_def
tff(fact_3949_old_Orec__nat__def,axiom,
    ! [T: $tType,X5: T,Xa: fun(nat,fun(T,T)),Xb: nat] : aa(nat,T,rec_nat(T,X5,Xa),Xb) = the(T,rec_set_nat(T,X5,Xa,Xb)) ).

% old.rec_nat_def
tff(fact_3950_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),comp(product_prod(A,B),A,product_prod(B,A),product_fst(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_oz(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_3951_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_pa(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_3952_effect__heapE,axiom,
    ! [A: $tType,F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_heap(A,F3),H,H5,R3,N)
     => ~ ( ( H5 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
         => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
           => ( R3 != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H)) ) ) ) ) ).

% effect_heapE
tff(fact_3953_effect__heapI,axiom,
    ! [A: $tType,N: nat,F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A] :
      ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
     => ( ( H5 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
       => ( ( R3 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H)) )
         => heap_Time_effect(A,heap_Time_heap(A,F3),H,H5,R3,N) ) ) ) ).

% effect_heapI
tff(fact_3954_effect__guardI,axiom,
    ! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),H5: heap_ext(product_unit),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),N: nat,R3: A] :
      ( pp(aa(heap_ext(product_unit),bool,P2,H))
     => ( ( H5 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
       => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
         => ( ( R3 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H)) )
           => heap_Time_effect(A,heap_Time_guard(A,P2,F3),H,H5,R3,N) ) ) ) ) ).

% effect_guardI
tff(fact_3955_effect__guardE,axiom,
    ! [A: $tType,P2: fun(heap_ext(product_unit),bool),F3: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_guard(A,P2,F3),H,H5,R3,N)
     => ~ ( ( H5 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
         => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H))) )
           => ( ( R3 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F3,H)) )
             => ~ pp(aa(heap_ext(product_unit),bool,P2,H)) ) ) ) ) ).

% effect_guardE
tff(fact_3956_ge__eq__refl,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R2))
     => pp(aa(A,bool,aa(A,fun(A,bool),R2,X),X)) ) ).

% ge_eq_refl
tff(fact_3957_refl__ge__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( ! [X3: A] : pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),X3))
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R2)) ) ).

% refl_ge_eq
tff(fact_3958_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2)) = the(product_prod(A,B),aTP_Lamp_pb(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P2)) ).

% The_case_prod
tff(fact_3959_range,axiom,
    ! [K2: code_natural,S2: product_prod(code_natural,code_natural)] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),K2))
     => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),code_natural,product_fst(code_natural,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),range(K2),S2))),K2)) ) ).

% range
tff(fact_3960_fstI,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),Y: A,Z4: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z4) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),X) = Y ) ) ).

% fstI
tff(fact_3961_sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),Y: A,Z4: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z4) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),X) = Z4 ) ) ).

% sndI
tff(fact_3962_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_pc(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_3963_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,nat),P3: product_prod(A,B)] : basic_BNF_size_prod(A,B,F3,G3,P3) = 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,F3,aa(product_prod(A,B),A,product_fst(A,B),P3))),aa(B,nat,G3,aa(product_prod(A,B),B,product_snd(A,B),P3)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_3964_the__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : the(A,aa(A,fun(A,bool),fequal(A),X)) = X ).

% the_sym_eq_trivial
tff(fact_3965_the__eq__trivial,axiom,
    ! [A: $tType,A3: A] : the(A,aTP_Lamp_ap(A,fun(A,bool),A3)) = A3 ).

% the_eq_trivial
tff(fact_3966_the__equality,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P2,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => ( X3 = A3 ) )
       => ( the(A,P2) = A3 ) ) ) ).

% the_equality
tff(fact_3967_theI,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P2,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => ( X3 = A3 ) )
       => pp(aa(A,bool,P2,the(A,P2))) ) ) ).

% theI
tff(fact_3968_theI_H,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P2,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P2,Y3))
             => ( Y3 = X5 ) ) )
     => pp(aa(A,bool,P2,the(A,P2))) ) ).

% theI'
tff(fact_3969_theI2,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A,Q2: fun(A,bool)] :
      ( pp(aa(A,bool,P2,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => ( X3 = A3 ) )
       => ( ! [X3: A] :
              ( pp(aa(A,bool,P2,X3))
             => pp(aa(A,bool,Q2,X3)) )
         => pp(aa(A,bool,Q2,the(A,P2))) ) ) ) ).

% theI2
tff(fact_3970_If__def,axiom,
    ! [A: $tType,P2: bool,X: A,Y: A] :
      ( ( pp(P2)
       => ( X = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_pd(bool,fun(A,fun(A,fun(A,bool))),P2),X),Y)) ) )
      & ( ~ pp(P2)
       => ( Y = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_pd(bool,fun(A,fun(A,fun(A,bool))),P2),X),Y)) ) ) ) ).

% If_def
tff(fact_3971_the1I2,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P2,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P2,Y3))
             => ( Y3 = X5 ) ) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => pp(aa(A,bool,Q2,X3)) )
       => pp(aa(A,bool,Q2,the(A,P2))) ) ) ).

% the1I2
tff(fact_3972_the1__equality,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P2,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P2,Y3))
             => ( Y3 = X5 ) ) )
     => ( pp(aa(A,bool,P2,A3))
       => ( the(A,P2) = A3 ) ) ) ).

% the1_equality
tff(fact_3973_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
     => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ).

% bezw_non_0
tff(fact_3974_times__int__def,axiom,
    times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nn(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_3975_pick_Osimps,axiom,
    ! [A: $tType,I2: code_natural,X: product_prod(code_natural,A),Xs: list(product_prod(code_natural,A))] :
      ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
       => ( aa(code_natural,A,pick(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),aa(product_prod(code_natural,A),fun(list(product_prod(code_natural,A)),list(product_prod(code_natural,A))),cons(product_prod(code_natural,A)),X),Xs)),I2) = aa(product_prod(code_natural,A),A,product_snd(code_natural,A),X) ) )
      & ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
       => ( aa(code_natural,A,pick(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),aa(product_prod(code_natural,A),fun(list(product_prod(code_natural,A)),list(product_prod(code_natural,A))),cons(product_prod(code_natural,A)),X),Xs)),I2) = aa(code_natural,A,pick(A,Xs),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),I2),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X))) ) ) ) ).

% pick.simps
tff(fact_3976_execute__bind__success,axiom,
    ! [B: $tType,A: $tType,F3: heap_Time_Heap(A),H: heap_ext(product_unit),G3: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_success(A,F3,H)
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F3,G3)),H) = heap_Time_timeFrame(B,aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H)))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G3,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H))))),aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F3),H)))))) ) ) ).

% execute_bind_success
tff(fact_3977_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_3978_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_3979_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_3980_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_3981_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A3: option(A),F3: fun(A,B)] :
      ( ( A3 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),map_option(A,B,F3),A3)) = aa(A,B,F3,aa(option(A),A,the2(A),A3)) ) ) ).

% option.map_sel
tff(fact_3982_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F1: B,F22: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( case_option(B,A,F1,F22,Option) = F1 ) )
      & ( ( Option != none(A) )
       => ( case_option(B,A,F1,F22,Option) = aa(A,B,F22,aa(option(A),A,the2(A),Option)) ) ) ) ).

% option.case_eq_if
tff(fact_3983_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P2: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P2,case_option(B,A,F1,F22,Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,P2,F1)) )
        & ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
         => pp(aa(B,bool,P2,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel
tff(fact_3984_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P2: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P2,case_option(B,A,F1,F22,Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,P2,F1)) )
          | ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
            & ~ pp(aa(B,bool,P2,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel_asm
tff(fact_3985_effectE,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,C3,H,H5,R3,N)
     => ~ ( ( R3 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H))) )
         => ( ( H5 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H)))) )
           => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C3),H)))) )
             => ~ heap_Time_success(A,C3,H) ) ) ) ) ).

% effectE
tff(fact_3986_plus__int__def,axiom,
    plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_3987_minus__int__def,axiom,
    minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ny(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_3988_normalize__def,axiom,
    ! [P3: product_prod(int,int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)))
       => ( normalize(P3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)))
       => ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int) )
           => ( normalize(P3) = 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(product_prod(int,int),int,product_snd(int,int),P3) != zero_zero(int) )
           => ( normalize(P3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))) ) ) ) ) ) ).

% normalize_def
tff(fact_3989_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs1: list(A),Xs22: list(A),Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs1)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),modulo_modulo(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))) )
      & ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs22)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_3990_wcount__add__mset,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,M6: multiset(A),A3: A] : wcount(A,F3,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),M6),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),X),A3),aa(nat,nat,suc,aa(A,nat,F3,A3)),zero_zero(nat))),wcount(A,F3,M6,A3)) ).

% wcount_add_mset
tff(fact_3991_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),X: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G3),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G3,aa(product_prod(C,D),D,product_snd(C,D),X))) ).

% apfst_apsnd
tff(fact_3992_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add1
tff(fact_3993_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add2
tff(fact_3994_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F3: fun(C,A),X: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X)),Y) ).

% apfst_conv
tff(fact_3995_apfst__eq__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),X: product_prod(C,B),G3: fun(C,A)] :
      ( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),X) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G3),X) )
    <=> ( aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),X)) = aa(C,A,G3,aa(product_prod(C,B),C,product_fst(C,B),X)) ) ) ).

% apfst_eq_conv
tff(fact_3996_fst__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),X: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),X)) = aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),X)) ).

% fst_apfst
tff(fact_3997_snd__apfst,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,B),X: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F3),X)) = aa(product_prod(C,A),A,product_snd(C,A),X) ).

% snd_apfst
tff(fact_3998_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)))
    <=> ( ( M != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_3999_snd__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),comp(product_prod(C,B),B,product_prod(A,B),product_snd(C,B)),product_apfst(A,C,B,F3)) = product_snd(A,B) ).

% snd_comp_apfst
tff(fact_4000_fst__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),comp(product_prod(C,B),C,product_prod(A,B),product_fst(C,B)),product_apfst(A,C,B,F3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_apfst
tff(fact_4001_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G3,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F3,aa(product_prod(D,C),C,product_snd(D,C),X))) ).

% apsnd_apfst
tff(fact_4002_apfst__compose,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,C),X: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G3),X)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),comp(C,A,D,F3),G3)),X) ).

% apfst_compose
tff(fact_4003_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs1: list(A),Xs22: list(A),X1: A,X2: A,Xs: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs))) = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs22)),Xs) ).

% mergesort_by_rel_split.simps(3)
tff(fact_4004_apsnd__apfst__commute,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),P3: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),P3)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G3),aa(product_prod(D,C),product_prod(D,B),aa(fun(C,B),fun(product_prod(D,C),product_prod(D,B)),product_apsnd(C,B,D),F3),P3)) ).

% apsnd_apfst_commute
tff(fact_4005_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K2: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K2),M)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add_mult
tff(fact_4006_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ).

% gcd_ge_0_int
tff(fact_4007_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V3: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U3),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V3),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% bezout_int
tff(fact_4008_gcd__le2__int,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),B2)) ) ).

% gcd_le2_int
tff(fact_4009_gcd__le1__int,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),A3)) ) ).

% gcd_le1_int
tff(fact_4010_gcd__cases__int,axiom,
    ! [X: int,Y: int,P2: fun(int,bool)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
         => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y)))) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
             => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y))) ) )
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y)))) ) )
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ) ) ).

% gcd_cases_int
tff(fact_4011_gcd__unique__int,axiom,
    ! [D3: int,A3: int,B2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D3))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),A3))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),B2))
        & ! [E4: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E4),A3))
              & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E4),B2)) )
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E4),D3)) ) )
    <=> ( D3 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2) ) ) ).

% gcd_unique_int
tff(fact_4012_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,X,Y)) ) ) ).

% gcd_non_0_int
tff(fact_4013_wcount__union,axiom,
    ! [A: $tType,F3: fun(A,nat),M6: multiset(A),N7: multiset(A),A3: A] : wcount(A,F3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),wcount(A,F3,M6,A3)),wcount(A,F3,N7,A3)) ).

% wcount_union
tff(fact_4014_part__code_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Pivot: A,X: B,Xs: list(B)] : linorder_part(B,A,F3,Pivot,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(product_prod(list(B),product_prod(list(B),list(B))),product_prod(list(B),product_prod(list(B),list(B))),aa(fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),fun(product_prod(list(B),product_prod(list(B),list(B))),product_prod(list(B),product_prod(list(B),list(B)))),product_case_prod(list(B),product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_pf(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),F3),Pivot),X)),linorder_part(B,A,F3,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_4015_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q5: product_prod(A,B),F3: fun(C,A),P3: product_prod(C,B)] :
      ( ( Q5 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),P3) )
     => ~ ! [X3: C,Y3: B] :
            ( ( P3 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Y3) )
           => ( Q5 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),Y3) ) ) ) ).

% apfst_convE
tff(fact_4016_mergesort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),P2: fun(fun(A,fun(A,bool)),fun(list(A),bool))] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),A0),A1)))
     => ( ! [R7: fun(A,fun(A,bool)),Xs2: list(A)] :
            ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R7),Xs2)))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
               => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P2,R7),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)))) )
             => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
                 => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P2,R7),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)))) )
               => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P2,R7),Xs2)) ) ) )
       => pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P2,A0),A1)) ) ) ).

% mergesort_by_rel.pinduct
tff(fact_4017_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
    <=> ( ( M != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_4018_slice__Nil,axiom,
    ! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).

% slice_Nil
tff(fact_4019_slice__eq__bounds__empty,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : slice(A,I2,I2,Xs) = nil(A) ).

% slice_eq_bounds_empty
tff(fact_4020_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_4021_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4022_mergesort__by__rel__merge__induct,axiom,
    ! [A: $tType,B: $tType,P2: fun(list(A),fun(list(B),bool)),R2: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( ! [Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs2),nil(B)))
     => ( ! [Ys5: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,nil(A)),Ys5))
       => ( ! [X3: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Y3))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5)))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5))) ) )
         => ( ! [X3: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
                ( ~ pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Y3))
               => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys5))
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5))) ) )
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs),Ys2)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4023_list__induct__first2,axiom,
    ! [A: $tType,P2: fun(list(A),bool),Xs: list(A)] :
      ( pp(aa(list(A),bool,P2,nil(A)))
     => ( ! [X3: A] : pp(aa(list(A),bool,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))
       => ( ! [X12: A,X22: A,Xs2: list(A)] :
              ( pp(aa(list(A),bool,P2,Xs2))
             => pp(aa(list(A),bool,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs2)))) )
         => pp(aa(list(A),bool,P2,Xs)) ) ) ) ).

% list_induct_first2
tff(fact_4024_list__2pre__induct,axiom,
    ! [A: $tType,B: $tType,P2: fun(list(A),fun(list(B),bool)),W1: list(A),W22: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,nil(A)),nil(B)))
     => ( ! [E2: A,W12: list(A),W23: list(B)] :
            ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,W12),W23))
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),W12)),W23)) )
       => ( ! [E2: B,W13: list(A),W24: list(B)] :
              ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,W13),W24))
             => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),W24))) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,W1),W22)) ) ) ) ).

% list_2pre_induct
tff(fact_4025_neq__NilE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [X3: A,Xs2: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) ) ).

% neq_NilE
tff(fact_4026_gcd__le2__nat,axiom,
    ! [B2: nat,A3: nat] :
      ( ( B2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2)) ) ).

% gcd_le2_nat
tff(fact_4027_gcd__le1__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),A3)) ) ).

% gcd_le1_nat
tff(fact_4028_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff1_nat
tff(fact_4029_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff2_nat
tff(fact_4030_len__greater__imp__nonempty,axiom,
    ! [A: $tType,X: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),L)))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_4031_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,X: product_prod(product_prod(list(A),list(A)),list(A))] :
      ( ! [Xs12: list(A),Xs23: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),nil(A))
     => ( ! [Xs12: list(A),Xs23: list(A),X3: A] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ~ ! [Xs12: list(A),Xs23: list(A),X12: A,X22: A,Xs2: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_4032_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
      ( ! [R7: fun(A,fun(A,bool)),X3: A,Xs2: list(A),Y3: A,Ys5: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))
     => ( ! [R7: fun(A,fun(A,bool)),Xs2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A)))
       => ~ ! [R7: fun(A,fun(A,bool)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_4033_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
      ( ! [R7: fun(A,fun(A,bool)),Sl: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
     => ~ ! [R7: fun(A,fun(A,bool)),Sl: list(A),X3: A,Xs2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_4034_partition__rev_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))] :
      ( ! [P: fun(A,bool),Yes: list(A),No: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
     => ~ ! [P: fun(A,bool),Yes: list(A),No: list(A),X3: A,Xs2: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ).

% partition_rev.cases
tff(fact_4035_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))] :
      ( ! [P: fun(A,fun(B,bool))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P),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)))
     => ( ! [P: fun(A,fun(B,bool)),A5: A,As2: list(A),B4: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2)))
       => ( ! [P: fun(A,fun(B,bool)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))
         => ~ ! [P: fun(A,fun(B,bool)),V3: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_4036_merge_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(A),list(A))] :
          ( ! [L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),L22)
         => ( ! [V3: A,Va: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(A))
           => ~ ! [X12: A,L1: list(A),X22: A,L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L22)) ) ) ) ).

% merge.cases
tff(fact_4037_zipf_Ocases,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
      ( ! [F: fun(A,fun(B,C))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F),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)))
     => ( ! [F: fun(A,fun(B,C)),A5: A,As2: list(A),B4: B,Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2)))
       => ( ! [A5: fun(A,fun(B,C)),V3: A,Va: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B)))
         => ~ ! [A5: fun(A,fun(B,C)),V3: B,Va: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))) ) ) ) ).

% zipf.cases
tff(fact_4038_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_4039_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_4040_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [A: $tType,Xs1: list(A),Xs22: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),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)),Xs1),Xs22) ).

% mergesort_by_rel_split.simps(1)
tff(fact_4041_bezout__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [X3: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) ) ).

% bezout_nat
tff(fact_4042_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A3: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) )
      | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_4043_length__compl__induct,axiom,
    ! [A: $tType,P2: fun(list(A),bool),L: list(A)] :
      ( pp(aa(list(A),bool,P2,nil(A)))
     => ( ! [E2: A,L3: list(A)] :
            ( ! [Ll: list(A)] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3)))
               => pp(aa(list(A),bool,P2,Ll)) )
           => pp(aa(list(A),bool,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L3))) )
       => pp(aa(list(A),bool,P2,L)) ) ) ).

% length_compl_induct
tff(fact_4044_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).

% sorted1
tff(fact_4045_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A5: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),nil(A)) ) ).

% list_decomp_1
tff(fact_4046_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs1: list(A),Xs22: list(A),X: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)),Xs22) ).

% mergesort_by_rel_split.simps(2)
tff(fact_4047_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A)),Xa2: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,X,Xa2) = Y )
     => ( ! [Xs12: list(A),Xs23: list(A)] :
            ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
           => ( ( Xa2 = nil(A) )
             => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) ) ) )
       => ( ! [Xs12: list(A),Xs23: list(A)] :
              ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
             => ! [X3: A] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                 => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs12)),Xs23) ) ) )
         => ~ ! [Xs12: list(A),Xs23: list(A)] :
                ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
               => ! [X12: A,X22: A,Xs2: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs2)) )
                   => ( Y != merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs23)),Xs2) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_4048_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_dt(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_4049_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)) )
     => ? [A5: A,B4: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B4),nil(A))) ) ).

% list_decomp_2
tff(fact_4050_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(X,Y))),aa(nat,int,semiring_1_of_nat(int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(X,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ).

% bezw_aux
tff(fact_4051_mergesort__by__rel__split_Opelims,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A)),Xa2: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),X),Xa2)))
       => ( ! [Xs12: list(A),Xs23: list(A)] :
              ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
             => ( ( Xa2 = nil(A) )
               => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
                 => ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),nil(A)))) ) ) )
         => ( ! [Xs12: list(A),Xs23: list(A)] :
                ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
               => ! [X3: A] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                   => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs12)),Xs23) )
                     => ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) ) )
           => ~ ! [Xs12: list(A),Xs23: list(A)] :
                  ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
                 => ! [X12: A,X22: A,Xs2: list(A)] :
                      ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs2)) )
                     => ( ( Y = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs23)),Xs2) )
                       => ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs2))))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.pelims
tff(fact_4052_mergesort__by__rel_Opsimps,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R2),Xs)))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
         => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = Xs ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
         => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = merges9089515139780605204_merge(A,R2,aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)))) ) ) ) ) ).

% mergesort_by_rel.psimps
tff(fact_4053_mergesort__by__rel_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa2) = Y )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa2)))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
               => ( Y = Xa2 ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
               => ( Y = merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2)))) ) ) )
           => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X),Xa2))) ) ) ) ).

% mergesort_by_rel.pelims
tff(fact_4054_upto_Opsimps,axiom,
    ! [I2: int,J2: int] :
      ( pp(aa(product_prod(int,int),bool,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)))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
         => ( upto(I2,J2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J2)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
         => ( upto(I2,J2) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_4055_mergesort__by__rel__simps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] : aa(list(A),list(A),mergesort_by_rel(A,R2),nil(A)) = nil(A) ).

% mergesort_by_rel_simps(1)
tff(fact_4056_set__mergesort__by__rel,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R2),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_mergesort_by_rel
tff(fact_4057_mergesort__by__rel__merge__simps_I3_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Ys2: list(A)] : merges9089515139780605204_merge(A,R2,nil(A),Ys2) = Ys2 ).

% mergesort_by_rel_merge_simps(3)
tff(fact_4058_sorted__wrt__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y3))
          | pp(aa(A,bool,aa(A,fun(A,bool),R2,Y3),X3)) )
     => ( ! [X3: A,Y3: A,Z3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,Y3),Z3))
             => pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Z3)) ) )
       => ( sorted_wrt(A,R2,merges9089515139780605204_merge(A,R2,Xs,Ys2))
        <=> ( sorted_wrt(A,R2,Xs)
            & sorted_wrt(A,R2,Ys2) ) ) ) ) ).

% sorted_wrt_mergesort_by_rel_merge
tff(fact_4059_mergesort__by__rel__simps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A] : aa(list(A),list(A),mergesort_by_rel(A,R2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).

% mergesort_by_rel_simps(2)
tff(fact_4060_set__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] : aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R2,Xs,Ys2)) = 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),Ys2)) ).

% set_mergesort_by_rel_merge
tff(fact_4061_upto__Nil,axiom,
    ! [I2: int,J2: int] :
      ( ( upto(I2,J2) = nil(int) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I2)) ) ).

% upto_Nil
tff(fact_4062_upto__Nil2,axiom,
    ! [I2: int,J2: int] :
      ( ( nil(int) = upto(I2,J2) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I2)) ) ).

% upto_Nil2
tff(fact_4063_upto__empty,axiom,
    ! [J2: int,I2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),I2))
     => ( upto(I2,J2) = nil(int) ) ) ).

% upto_empty
tff(fact_4064_nth__upto,axiom,
    ! [I2: int,K2: nat,J2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K2))),J2))
     => ( aa(nat,int,nth(int,upto(I2,J2)),K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K2)) ) ) ).

% nth_upto
tff(fact_4065_length__upto,axiom,
    ! [I2: int,J2: int] : aa(list(int),nat,size_size(list(int)),upto(I2,J2)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),I2)),one_one(int))) ).

% length_upto
tff(fact_4066_mergesort__by__rel__simps_I3_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X1: A,X2: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs))) = aa(product_prod(list(A),list(A)),list(A),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)),aTP_Lamp_pg(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),R2)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),Xs)) ).

% mergesort_by_rel_simps(3)
tff(fact_4067_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(1)
tff(fact_4068_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(4)
tff(fact_4069_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(3)
tff(fact_4070_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(2)
tff(fact_4071_sorted__wrt__upto,axiom,
    ! [I2: int,J2: int] : sorted_wrt(int,ord_less(int),upto(I2,J2)) ).

% sorted_wrt_upto
tff(fact_4072_sorted__upto,axiom,
    ! [M: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M,N)) ).

% sorted_upto
tff(fact_4073_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),X: A,Y: A,Xs: list(A),Ys2: list(A)] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y))
       => ( merges9089515139780605204_merge(A,R2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,R2,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y))
       => ( merges9089515139780605204_merge(A,R2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2)) ) ) ) ).

% mergesort_by_rel_merge_simps(1)
tff(fact_4074_mergesort__by__rel__merge__simps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] : merges9089515139780605204_merge(A,R2,Xs,nil(A)) = Xs ).

% mergesort_by_rel_merge_simps(2)
tff(fact_4075_sorted__wrt__mergesort__by__rel,axiom,
    ! [X9: $tType,R2: fun(X9,fun(X9,bool)),Xs: list(X9)] :
      ( ! [X3: X9,Y3: X9] :
          ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,X3),Y3))
          | pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,Y3),X3)) )
     => ( ! [X3: X9,Y3: X9,Z3: X9] :
            ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,X3),Y3))
           => ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,Y3),Z3))
             => pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,X3),Z3)) ) )
       => sorted_wrt(X9,R2,aa(list(X9),list(X9),mergesort_by_rel(X9,R2),Xs)) ) ) ).

% sorted_wrt_mergesort_by_rel
tff(fact_4076_sorted__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_by_rel(A,ord_less_eq(A)),Xs)) ) ).

% sorted_mergesort_by_rel
tff(fact_4077_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),V2: A,Va2: list(A)] : merges9089515139780605204_merge(A,R2,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ).

% mergesort_by_rel_merge.simps(3)
tff(fact_4078_mergesort__by__rel__merge_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb2: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,X,Xa2,Xb2) = Y )
     => ( ! [X3: A,Xs2: list(A)] :
            ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
           => ! [Y3: A,Ys5: list(A)] :
                ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
               => ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                     => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5))) ) )
                    & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                     => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),merges9089515139780605204_merge(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys5)) ) ) ) ) )
       => ( ( ( Xb2 = nil(A) )
           => ( Y != Xa2 ) )
         => ~ ( ( Xa2 = nil(A) )
             => ! [V3: A,Va: list(A)] :
                  ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.elims
tff(fact_4079_greaterThanAtMost__upto,axiom,
    ! [I2: int,J2: int] : set_or3652927894154168847AtMost(int,I2,J2) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J2)) ).

% greaterThanAtMost_upto
tff(fact_4080_upto__rec1,axiom,
    ! [I2: int,J2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
     => ( upto(I2,J2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J2)) ) ) ).

% upto_rec1
tff(fact_4081_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
         => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
         => ( Y = nil(int) ) ) ) ) ).

% upto.elims
tff(fact_4082_upto_Osimps,axiom,
    ! [I2: int,J2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
       => ( upto(I2,J2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J2)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
       => ( upto(I2,J2) = nil(int) ) ) ) ).

% upto.simps
tff(fact_4083_greaterThanLessThan__upto,axiom,
    ! [I2: int,J2: int] : set_or5935395276787703475ssThan(int,I2,J2) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_4084_mergesort__by__rel_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,X),Xa2) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
         => ( Y = Xa2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
         => ( Y = merges9089515139780605204_merge(A,X,aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2))),aa(list(A),list(A),mergesort_by_rel(A,X),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa2)))) ) ) ) ) ).

% mergesort_by_rel.elims
tff(fact_4085_mergesort__by__rel_Osimps,axiom,
    ! [A: $tType,Xs: list(A),R2: fun(A,fun(A,bool))] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
       => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = Xs ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))
       => ( aa(list(A),list(A),mergesort_by_rel(A,R2),Xs) = merges9089515139780605204_merge(A,R2,aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R2),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)))) ) ) ) ).

% mergesort_by_rel.simps
tff(fact_4086_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( pp(aa(product_prod(int,int),bool,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),X),Xa2)))
       => ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
               => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
               => ( Y = nil(int) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,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),X),Xa2))) ) ) ) ).

% upto.pelims
tff(fact_4087_mergesort__by__rel__merge_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb2: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,X,Xa2,Xb2) = Y )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),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)),Xa2),Xb2))))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Y3: A,Ys5: list(A)] :
                  ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                 => ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                       => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5))) ) )
                      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3))
                       => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),merges9089515139780605204_merge(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys5)) ) ) )
                   => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5))))) ) ) )
         => ( ( ( Xb2 = nil(A) )
             => ( ( Y = Xa2 )
               => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),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)),Xa2),nil(A))))) ) )
           => ~ ( ( Xa2 = nil(A) )
               => ! [V3: A,Va: list(A)] :
                    ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                   => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                     => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va))))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
tff(fact_4088_mergesort__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% mergesort_def
tff(fact_4089_extract__Cons__code,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P2,X))
       => ( extract(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),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)),X),Xs))) ) )
      & ( ~ pp(aa(A,bool,P2,X))
       => ( extract(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),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_pi(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X)),extract(A,P2,Xs)) ) ) ) ).

% extract_Cons_code
tff(fact_4090_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),N: A,Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns))),lenlex(A,R3)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N)),R3)) )
        | ( ( M = N )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R3))) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4091_lenlex__irreflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lenlex(A,R3))) ) ).

% lenlex_irreflexive
tff(fact_4092_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R3)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).

% lenlex_length
tff(fact_4093_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ~ ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),S))
                  & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X5)),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S)))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_4094_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),lex(A,R3)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),lex(A,R3))) ) ) ) ).

% Cons_in_lex
tff(fact_4095_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Y: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,member(A,Y),S))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S))),aa(A,B,F3,Y))) ) ) ) ) ).

% arg_min_least
tff(fact_4096_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_4097_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_4098_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
     => ( num_of_nat(N) = one ) ) ).

% num_of_nat_One
tff(fact_4099_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => pp(aa(set(A),bool,member(A,lattic7623131987881927897min_on(A,B,F3,S)),S)) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_4100_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = aa(num,num,bit0,num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_4101_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_4102_lenlex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pj(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lenlex_conv
tff(fact_4103_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4104_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,one) ) ).

% add_neg_numeral_special(4)
tff(fact_4105_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M,one) ) ).

% add_neg_numeral_special(3)
tff(fact_4106_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = neg_numeral_sub(A,one,M) ) ).

% add_neg_numeral_special(2)
tff(fact_4107_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V2,W2)),Y) ) ).

% semiring_norm(166)
tff(fact_4108_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W2,V2)),Y) ) ).

% semiring_norm(167)
tff(fact_4109_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_4110_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_4111_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A6)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4112_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) = neg_numeral_sub(A,one,M) ) ).

% add_neg_numeral_special(1)
tff(fact_4113_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: 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_pk(A,A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X),Xs)) ) ).

% insort_left_comm
tff(fact_4114_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4115_insort__key_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Y: B,Ys2: list(B)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Ys2)) ) ) ) ) ).

% insort_key.simps(2)
tff(fact_4116_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: 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_pk(A,A)),X),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_4117_insort__is__Cons,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),F3: fun(B,A),A3: B] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),aa(list(B),set(B),set2(B),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,A3)),aa(B,A,F3,X3))) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),A3),Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_4118_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).

% sub_non_positive
tff(fact_4119_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% sub_non_negative
tff(fact_4120_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% sub_positive
tff(fact_4121_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).

% sub_negative
tff(fact_4122_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A6) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4123_sorted__list__of__multiset__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,M6: multiset(A)] : linord6283353356039996273ltiset(A,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),M6)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X),linord6283353356039996273ltiset(A,M6)) ) ).

% sorted_list_of_multiset_insert
tff(fact_4124_pred__nat__def,axiom,
    pred_nat = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_pl(nat,fun(nat,bool)))) ).

% pred_nat_def
tff(fact_4125_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4126_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4127_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4128_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4129_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N)),transitive_trancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% less_eq
tff(fact_4130_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P3: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P3) )
    <=> ? [A9: B] : P3 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A9),B2) ) ).

% eq_snd_iff
tff(fact_4131_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,P3: product_prod(A,B)] :
      ( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P3) )
    <=> ? [B7: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B7) ) ).

% eq_fst_iff
tff(fact_4132_merge__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L12: list(A),L23: list(A)] :
          ( ( distinct(A,L12)
            & sorted_wrt(A,ord_less_eq(A),L12) )
         => ( ( distinct(A,L23)
              & sorted_wrt(A,ord_less_eq(A),L23) )
           => ( distinct(A,merge(A,L12,L23))
              & sorted_wrt(A,ord_less_eq(A),merge(A,L12,L23))
              & ( aa(list(A),set(A),set2(A),merge(A,L12,L23)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),L12)),aa(list(A),set(A),set2(A),L23)) ) ) ) ) ) ).

% merge_correct
tff(fact_4133_distinct__foldl__invar,axiom,
    ! [B: $tType,A: $tType,S: list(A),I5: fun(set(A),fun(B,bool)),Sigma_0: B,F3: fun(B,fun(A,B))] :
      ( distinct(A,S)
     => ( pp(aa(B,bool,aa(set(A),fun(B,bool),I5,aa(list(A),set(A),set2(A),S)),Sigma_0))
       => ( ! [X3: A,It: set(A),Sigma: B] :
              ( pp(aa(set(A),bool,member(A,X3),It))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S)))
               => ( pp(aa(B,bool,aa(set(A),fun(B,bool),I5,It),Sigma))
                 => pp(aa(B,bool,aa(set(A),fun(B,bool),I5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),It),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F3,Sigma),X3))) ) ) )
         => pp(aa(B,bool,aa(set(A),fun(B,bool),I5,bot_bot(set(A))),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),Sigma_0),S))) ) ) ) ).

% distinct_foldl_invar
tff(fact_4134_foldl__length,axiom,
    ! [A: $tType,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_pm(nat,fun(A,nat))),zero_zero(nat)),L) = aa(list(A),nat,size_size(list(A)),L) ).

% foldl_length
tff(fact_4135_foldl__A1__eq,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),N: A,I2: A,Ww: list(A)] :
      ( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F3,N),E2) = E2
     => ( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F3,E2),N) = E2
       => ( ! [A5: A,B4: A,C2: A] : aa(A,A,aa(A,fun(A,A),F3,A5),aa(A,A,aa(A,fun(A,A),F3,B4),C2)) = aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,A5),B4)),C2)
         => ( aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F3),I2),Ww) = aa(A,A,aa(A,fun(A,A),F3,I2),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F3),N),Ww)) ) ) ) ) ).

% foldl_A1_eq
tff(fact_4136_merge_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L23: list(A)] : merge(A,nil(A),L23) = L23 ) ).

% merge.simps(1)
tff(fact_4137_fst__foldl,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,fun(C,A)),G3: fun(A,fun(B,fun(C,B))),A3: A,B2: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(list(C),product_prod(A,B),aa(product_prod(A,B),fun(list(C),product_prod(A,B)),foldl(product_prod(A,B),C,aa(fun(A,fun(B,fun(C,product_prod(A,B)))),fun(product_prod(A,B),fun(C,product_prod(A,B))),product_case_prod(A,B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_pn(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F3),G3))),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),foldl(A,C,F3),A3),Xs) ).

% fst_foldl
tff(fact_4138_foldl__absorb1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Zs: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),one_one(A)),Zs)) = aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),X),Zs) ) ).

% foldl_absorb1
tff(fact_4139_foldl__un__empty__eq,axiom,
    ! [A: $tType,I2: set(A),Ww: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),I2),Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I2),aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),bot_bot(set(A))),Ww)) ).

% foldl_un_empty_eq
tff(fact_4140_foldl__snd__zip,axiom,
    ! [B: $tType,C: $tType,A: $tType,Ys2: list(A),Xs: list(B),F3: fun(C,fun(A,C)),B2: C] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(B),nat,size_size(list(B)),Xs)))
     => ( aa(list(product_prod(B,A)),C,aa(C,fun(list(product_prod(B,A)),C),foldl(C,product_prod(B,A),aTP_Lamp_pp(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F3)),B2),zip(B,A,Xs,Ys2)) = aa(list(A),C,aa(C,fun(list(A),C),foldl(C,A,F3),B2),Ys2) ) ) ).

% foldl_snd_zip
tff(fact_4141_merge_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X1: A,X2: A,L12: list(A),L23: list(A)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X2))
           => ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L23))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X2))
           => ( ( ( X1 = X2 )
               => ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L12,L23)) ) )
              & ( ( X1 != X2 )
               => ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L12),L23)) ) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_4142_merge_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [V2: A,Va2: list(A)] : merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Va2) ) ).

% merge.simps(2)
tff(fact_4143_foldl__length__aux,axiom,
    ! [A: $tType,A3: nat,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_pm(nat,fun(A,nat))),A3),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).

% foldl_length_aux
tff(fact_4144_merge_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Xa2: list(A),Y: list(A)] :
          ( ( merge(A,X,Xa2) = Y )
         => ( ( ( X = nil(A) )
             => ( Y != Xa2 ) )
           => ( ! [V3: A,Va: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( ( Xa2 = nil(A) )
                   => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) ) ) )
             => ~ ! [X12: A,L1: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1) )
                   => ! [X22: A,L22: list(A)] :
                        ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L22) )
                       => ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                             => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L22))) ) )
                            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                             => ( ( ( X12 = X22 )
                                 => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,L22)) ) )
                                & ( ( X12 != X22 )
                                 => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1),L22)) ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
tff(fact_4145_merge_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Xa2: list(A),Y: list(A)] :
          ( ( merge(A,X,Xa2) = Y )
         => ( pp(aa(product_prod(list(A),list(A)),bool,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)),X),Xa2)))
           => ( ( ( X = nil(A) )
               => ( ( Y = Xa2 )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,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)),Xa2))) ) )
             => ( ! [V3: A,Va: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                   => ( ( Xa2 = nil(A) )
                     => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                       => ~ pp(aa(product_prod(list(A),list(A)),bool,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),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(A)))) ) ) )
               => ~ ! [X12: A,L1: list(A)] :
                      ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1) )
                     => ! [X22: A,L22: list(A)] :
                          ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L22) )
                         => ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                               => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L22))) ) )
                              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X22))
                               => ( ( ( X12 = X22 )
                                   => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L1,L22)) ) )
                                  & ( ( X12 != X22 )
                                   => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L1),L22)) ) ) ) ) )
                           => ~ pp(aa(product_prod(list(A),list(A)),bool,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),aa(A,fun(list(A),list(A)),cons(A),X12),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L22)))) ) ) ) ) ) ) ) ) ).

% merge.pelims
tff(fact_4146_rat__floor__lemma,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),one_one(int))))) ) ).

% rat_floor_lemma
tff(fact_4147_upto__rec2,axiom,
    ! [I2: int,J2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
     => ( upto(I2,J2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J2),nil(int))) ) ) ).

% upto_rec2
tff(fact_4148_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
         => ( ( pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1962203154675924110t_prod(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = groups1962203154675924110t_prod(B,A,P3,I5) ) )
            & ( ~ pp(aa(set(B),bool,member(B,I2),I5))
             => ( groups1962203154675924110t_prod(B,A,P3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P3,I2)),groups1962203154675924110t_prod(B,A,P3,I5)) ) ) ) ) ) ).

% prod.insert'
tff(fact_4149_empty__append__eq__id,axiom,
    ! [A: $tType,X5: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),X5) = X5 ).

% empty_append_eq_id
tff(fact_4150_list__e__eq__lel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E5: A,L23: list(A),E3: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) )
    <=> ( ( L12 = nil(A) )
        & ( E5 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(2)
tff(fact_4151_list__e__eq__lel_I1_J,axiom,
    ! [A: $tType,E3: A,L12: list(A),E5: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) )
    <=> ( ( L12 = nil(A) )
        & ( E5 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(1)
tff(fact_4152_list__se__match_I4_J,axiom,
    ! [A: $tType,L23: list(A),A3: A,L12: list(A)] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).

% list_se_match(4)
tff(fact_4153_list__se__match_I3_J,axiom,
    ! [A: $tType,L12: list(A),A3: A,L23: list(A)] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(3)
tff(fact_4154_list__se__match_I2_J,axiom,
    ! [A: $tType,L23: list(A),L12: list(A),A3: A] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).

% list_se_match(2)
tff(fact_4155_list__se__match_I1_J,axiom,
    ! [A: $tType,L12: list(A),L23: list(A),A3: A] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(1)
tff(fact_4156_list__ee__eq__leel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E1: A,E22: A,L23: list(A),E12: A,E23: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),L23))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),nil(A))) )
    <=> ( ( L12 = nil(A) )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_4157_list__ee__eq__leel_I1_J,axiom,
    ! [A: $tType,E12: A,E23: A,L12: list(A),E1: A,E22: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),L23))) )
    <=> ( ( L12 = nil(A) )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_4158_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = 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)),Ys2)) ).

% length_append
tff(fact_4159_nth__append__first,axiom,
    ! [A: $tType,I2: nat,L: list(A),L4: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L4)),I2) = aa(nat,A,nth(A,L),I2) ) ) ).

% nth_append_first
tff(fact_4160_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys2),N) ).

% nth_append_length_plus
tff(fact_4161_less__rat,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C3),D3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)))) ) ) ) ).

% less_rat
tff(fact_4162_le__rat,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C3),D3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)))) ) ) ) ).

% le_rat
tff(fact_4163_add__rat,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C3),D3)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).

% add_rat
tff(fact_4164_foldl__conc__empty__eq,axiom,
    ! [A: $tType,I2: list(A),Ww: list(list(A))] : aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),I2),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),I2),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),nil(A)),Ww)) ).

% foldl_conc_empty_eq
tff(fact_4165_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(list(A)),list(list(A)))] :
          ( ( X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
         => ( ! [L3: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))
           => ( ! [La: list(A),Acc22: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))
             => ( ! [La: list(A),Acc22: list(list(A)),L3: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))
               => ~ ! [Acc22: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))) ) ) ) ) ) ).

% merge_list.cases
tff(fact_4166_list__match__lel__lel,axiom,
    ! [A: $tType,C1: list(A),Qs: A,C22: list(A),C12: list(A),Qs2: A,C23: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) )
     => ( ! [C21: list(A)] :
            ( ( C1 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
           => ( C23 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C21),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list(A)] :
                ( ( C12 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C212)) )
               => ( C22 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C212),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_4167_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A)] : enumerate(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),append(product_prod(nat,A)),enumerate(A,N,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys2)) ).

% enumerate_append_eq
tff(fact_4168_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),I5: set(B)] : groups1962203154675924110t_prod(B,A,G3,aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_pq(fun(B,A),fun(set(B),fun(B,bool)),G3),I5))) = groups1962203154675924110t_prod(B,A,G3,I5) ) ).

% prod.non_neutral'
tff(fact_4169_neq__Nil__revE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [Ll2: list(A),E2: A] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ll2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A))) ) ).

% neq_Nil_revE
tff(fact_4170_rev__induct2_H,axiom,
    ! [A: $tType,B: $tType,P2: fun(list(A),fun(list(B),bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,nil(A)),nil(B)))
     => ( ! [X3: A,Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),nil(B)))
       => ( ! [Y3: B,Ys5: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,nil(A)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))))
         => ( ! [X3: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
                ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs2),Ys5))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))) )
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs),Ys2)) ) ) ) ) ).

% rev_induct2'
tff(fact_4171_neq__Nil__rev__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
    <=> ? [Xs3: list(A),X4: A] : L = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))) ) ).

% neq_Nil_rev_conv
tff(fact_4172_rev__nonempty__induct2_H,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),P2: fun(list(A),fun(list(B),bool))] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(B) )
       => ( ! [X3: A,Y3: B] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))
         => ( ! [X3: A,Xs2: list(A),Y3: B] :
                ( ( Xs2 != nil(A) )
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))) )
           => ( ! [X3: A,Y3: B,Ys5: list(B)] :
                  ( ( Ys5 != nil(B) )
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))) )
             => ( ! [X3: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
                    ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs2),Ys5))
                   => ( ( Xs2 != nil(A) )
                     => ( ( Ys5 != nil(B) )
                       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))) ) ) )
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs),Ys2)) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_4173_list__Cons__eq__append__cases,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Zs) )
     => ( ( ( Ys2 = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
       => ~ ! [Ys3: list(A)] :
              ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys3) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_4174_list__append__eq__Cons__cases,axiom,
    ! [A: $tType,Ys2: list(A),Zs: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
     => ( ( ( Ys2 = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
       => ~ ! [Ys3: list(A)] :
              ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys3) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_4175_Rat__induct__pos,axiom,
    ! [P2: fun(rat,bool),Q5: rat] :
      ( ! [A5: int,B4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
         => pp(aa(rat,bool,P2,aa(int,rat,aa(int,fun(int,rat),fract,A5),B4))) )
     => pp(aa(rat,bool,P2,Q5)) ) ).

% Rat_induct_pos
tff(fact_4176_xy__in__set__cases,axiom,
    ! [A: $tType,X: A,L: list(A),Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L)))
     => ( pp(aa(set(A),bool,member(A,Y),aa(list(A),set(A),set2(A),L)))
       => ( ( ( X = Y )
           => ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L22)) )
         => ( ( ( X != Y )
             => ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L32)))) )
           => ~ ( ( X != Y )
               => ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L32)))) ) ) ) ) ) ).

% xy_in_set_cases
tff(fact_4177_in__set__list__format,axiom,
    ! [A: $tType,E3: A,L: list(A)] :
      ( pp(aa(set(A),bool,member(A,E3),aa(list(A),set(A),set2(A),L)))
     => ~ ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),L22)) ) ).

% in_set_list_format
tff(fact_4178_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R12: list(A),R23: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1)))
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S1),R12) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S22),R23) )
       => ? [R1p: list(A)] : R23 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),R1p),R12) ) ) ).

% list_rest_coinc
tff(fact_4179_set__union__code,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : 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),Ys2)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) ).

% set_union_code
tff(fact_4180_distinct__match,axiom,
    ! [A: $tType,Al: list(A),E3: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)))
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl2)) )
      <=> ( ( Al = Al2 )
          & ( Bl = Bl2 ) ) ) ) ).

% distinct_match
tff(fact_4181_prod_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ij(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G3,I5)),groups1962203154675924110t_prod(B,A,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_4182_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys2)
            & ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
               => ! [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),aa(list(A),set(A),set2(A),Ys2)))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa3)) ) ) ) ) ) ).

% sorted_append
tff(fact_4183_list__update__append1,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Ys2: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2),I2,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,I2,X)),Ys2) ) ) ).

% list_update_append1
tff(fact_4184_foldl__rule,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2))] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma_0),nil(A)),L0))
     => ( ! [L1: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),L22)) ) )
       => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0)),L0),nil(A))) ) ) ).

% foldl_rule
tff(fact_4185_foldl__rule__P,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2)),P2: fun(Sigma2,bool)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma_0),nil(A)),L0))
     => ( ! [L1: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),L22)) ) )
       => ( ! [Sigma: Sigma2] :
              ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I5,Sigma),L0),nil(A)))
             => pp(aa(Sigma2,bool,P2,Sigma)) )
         => pp(aa(Sigma2,bool,P2,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0))) ) ) ) ).

% foldl_rule_P
tff(fact_4186_foldl__rule__aux,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2))] :
      ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma_0),L0))
     => ( ! [L1: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
           => ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
             => pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),L22)) ) )
       => pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0)),nil(A))) ) ) ).

% foldl_rule_aux
tff(fact_4187_foldl__rule__aux__P,axiom,
    ! [Sigma2: $tType,A: $tType,I5: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F3: fun(Sigma2,fun(A,Sigma2)),P2: fun(Sigma2,bool)] :
      ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma_0),L0))
     => ( ! [L1: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
            ( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
           => ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
             => pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F3,Sigma),X3)),L22)) ) )
       => ( ! [Sigma: Sigma2] :
              ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I5,Sigma),nil(A)))
             => pp(aa(Sigma2,bool,P2,Sigma)) )
         => pp(aa(Sigma2,bool,P2,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F3),Sigma_0),L0))) ) ) ) ).

% foldl_rule_aux_P
tff(fact_4188_lex__append__left__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,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)),Ys2),Zs)),lex(A,R3))) ) ) ).

% lex_append_left_iff
tff(fact_4189_lex__append__leftD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,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)),Ys2),Zs)),lex(A,R3))) ) ) ).

% lex_append_leftD
tff(fact_4190_prod_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T8: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,G3,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G3,S) = groups1962203154675924110t_prod(B,A,G3,T8) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_4191_prod_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T8: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,G3,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G3,T8) = groups1962203154675924110t_prod(B,A,G3,S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_4192_prod_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T8: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,H,I3) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G3,S) = groups1962203154675924110t_prod(B,A,H,T8) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_4193_prod_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T8: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S),T8))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T8),S)))
               => ( aa(B,A,G3,X3) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),S))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G3,T8) = groups1962203154675924110t_prod(B,A,H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_4194_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Ys4: list(A),Y4: A] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_4195_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_4196_length__compl__rev__induct,axiom,
    ! [A: $tType,P2: fun(list(A),bool),L: list(A)] :
      ( pp(aa(list(A),bool,P2,nil(A)))
     => ( ! [L3: list(A),E2: A] :
            ( ! [Ll: list(A)] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L3)))
               => pp(aa(list(A),bool,P2,Ll)) )
           => pp(aa(list(A),bool,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A))))) )
       => pp(aa(list(A),bool,P2,L)) ) ) ).

% length_compl_rev_induct
tff(fact_4197_not__distinct__split__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ~ distinct(A,Xs)
     => ~ ! [Y3: A,Ys5: list(A)] :
            ( distinct(A,Ys5)
           => ( pp(aa(set(A),bool,member(A,Y3),aa(list(A),set(A),set2(A),Ys5)))
             => ! [Zs2: list(A)] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))),Zs2)) ) ) ) ).

% not_distinct_split_distinct
tff(fact_4198_nth__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),N) = aa(nat,A,nth(A,Xs),N) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),N) = aa(nat,A,nth(A,Ys2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).

% nth_append
tff(fact_4199_list__update__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys2: list(A),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2),N,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,N,X)),Ys2) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2),N,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),list_update(A,Ys2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),X)) ) ) ) ).

% list_update_append
tff(fact_4200_prod_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),G3))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ij(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G3,I5)),groups1962203154675924110t_prod(B,A,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_4201_upto__split2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),K2))
       => ( upto(I2,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J2)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J2),one_one(int)),K2)) ) ) ) ).

% upto_split2
tff(fact_4202_upto__split1,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),K2))
       => ( upto(I2,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)))),upto(J2,K2)) ) ) ) ).

% upto_split1
tff(fact_4203_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P3: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1962203154675924110t_prod(B,A,P3,I5) = groups7121269368397514597t_prod(B,A,P3,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),I5),P3))))
           => ( groups1962203154675924110t_prod(B,A,P3,I5) = one_one(A) ) ) ) ) ).

% prod.G_def
tff(fact_4204_zero__less__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).

% zero_less_Fract_iff
tff(fact_4205_Fract__less__zero__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).

% Fract_less_zero_iff
tff(fact_4206_Fract__less__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B2)) ) ) ).

% Fract_less_one_iff
tff(fact_4207_one__less__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A3)) ) ) ).

% one_less_Fract_iff
tff(fact_4208_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)),N) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,M),N)),one_one(rat)) ) ) ).

% Fract_add_one
tff(fact_4209_slice__prepend,axiom,
    ! [A: $tType,I2: nat,K2: nat,Xs: list(A),Ys2: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( slice(A,I2,K2,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(list(A),nat,size_size(list(A)),Ys2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Xs)) ) ) ) ).

% slice_prepend
tff(fact_4210_sorted__append__bigger,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y)) )
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ) ) ) ).

% sorted_append_bigger
tff(fact_4211_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B),Ys2: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Ys2))) ) ).

% horner_sum_append
tff(fact_4212_Fract__le__zero__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).

% Fract_le_zero_iff
tff(fact_4213_zero__le__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).

% zero_le_Fract_iff
tff(fact_4214_one__le__Fract__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3)) ) ) ).

% one_le_Fract_iff
tff(fact_4215_Fract__le__one__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B2)) ) ) ).

% Fract_le_one_iff
tff(fact_4216_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),A3),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_4217_upto__split3,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),K2))
       => ( upto(I2,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J2),one_one(int)),K2))) ) ) ) ).

% upto_split3
tff(fact_4218_subset__eq__mset__impl_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: option(bool)] :
      ( ( subset_eq_mset_impl(A,X,Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa2),nil(A)))) ) )
       => ~ ! [X3: A,Xs2: list(A)] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ( Y != case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_ps(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2)),extract(A,aa(A,fun(A,bool),fequal(A),X3),Xa2)) ) ) ) ) ).

% subset_eq_mset_impl.elims
tff(fact_4219_subset__eq__mset__impl_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: option(bool)] :
      ( ( subset_eq_mset_impl(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,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)),X),Xa2)))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa2),nil(A)))) )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,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)),Xa2))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( ( Y = case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_ps(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2)),extract(A,aa(A,fun(A,bool),fequal(A),X3),Xa2)) )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xa2))) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
tff(fact_4220_Succ__def,axiom,
    ! [A: $tType,Kl: set(list(A)),Kl2: list(A)] : bNF_Greatest_Succ(A,Kl,Kl2) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_pt(set(list(A)),fun(list(A),fun(A,bool)),Kl),Kl2)) ).

% Succ_def
tff(fact_4221_sort__key__by__quicksort__code,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = aa(list(B),list(B),case_list(list(B),B,nil(B),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_py(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),F3),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_4222_sort__upto,axiom,
    ! [I2: int,J2: int] : aa(list(int),list(int),linorder_sort_key(int,int,aTP_Lamp_ia(int,int)),upto(I2,J2)) = upto(I2,J2) ).

% sort_upto
tff(fact_4223_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C3: B,Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,aTP_Lamp_pz(B,fun(A,B),C3)),Xs) = Xs ) ).

% sort_key_const
tff(fact_4224_list_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,fun(list(A),B)),List: list(A)] : aa(B,C,H,aa(list(A),B,case_list(B,A,F1,F22),List)) = aa(list(A),C,case_list(C,A,aa(B,C,H,F1),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_qa(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H),F22)),List) ).

% list.case_distrib
tff(fact_4225_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_pk(A,A)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_4226_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_pk(A,A)),Xs)) ) ).

% sorted_sort
tff(fact_4227_sort__mergesort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_pk(A,A)) = mergesort(A) ) ) ).

% sort_mergesort
tff(fact_4228_sort__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_pk(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% sort_mergesort_by_rel
tff(fact_4229_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys2: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2) = 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_qb(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys2) ).

% zip_Cons1
tff(fact_4230_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys2: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2)) = 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_qc(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys2)),Xs) ).

% zip_Cons
tff(fact_4231_SuccI,axiom,
    ! [A: $tType,Kl2: list(A),K2: A,Kl: set(list(A))] :
      ( pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Kl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K2),nil(A)))),Kl))
     => pp(aa(set(A),bool,member(A,K2),bNF_Greatest_Succ(A,Kl,Kl2))) ) ).

% SuccI
tff(fact_4232_SuccD,axiom,
    ! [A: $tType,K2: A,Kl: set(list(A)),Kl2: list(A)] :
      ( pp(aa(set(A),bool,member(A,K2),bNF_Greatest_Succ(A,Kl,Kl2)))
     => pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Kl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K2),nil(A)))),Kl)) ) ).

% SuccD
tff(fact_4233_subset__eq__mset__impl_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A)] : subset_eq_mset_impl(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2) = case_option(option(bool),product_prod(list(A),product_prod(A,list(A))),none(bool),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_ps(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs)),extract(A,aa(A,fun(A,bool),fequal(A),X),Ys2)) ).

% subset_eq_mset_impl.simps(2)
tff(fact_4234_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),filter2(A,aTP_Lamp_qd(list(A),fun(A,bool),Xs),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),filter2(A,aTP_Lamp_qe(list(A),fun(A,bool),Xs),Xs)),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),filter2(A,aa(A,fun(A,bool),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),aa(num,num,bit0,one))))),Xs)))) ) ).

% sort_by_quicksort
tff(fact_4235_sort__key__by__quicksort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_sort_key(B,A,F3),filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_qf(fun(B,A),fun(list(B),fun(B,bool)),F3),Xs),Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_qg(fun(B,A),fun(list(B),fun(B,bool)),F3),Xs),Xs)),aa(list(B),list(B),linorder_sort_key(B,A,F3),filter2(B,aa(list(B),fun(B,bool),aTP_Lamp_qh(fun(B,A),fun(list(B),fun(B,bool)),F3),Xs),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_4236_empty__Shift,axiom,
    ! [A: $tType,Kl: set(list(A)),K2: A] :
      ( pp(aa(set(list(A)),bool,member(list(A),nil(A)),Kl))
     => ( pp(aa(set(A),bool,member(A,K2),bNF_Greatest_Succ(A,Kl,nil(A))))
       => pp(aa(set(list(A)),bool,member(list(A),nil(A)),bNF_Greatest_Shift(A,Kl,K2))) ) ) ).

% empty_Shift
tff(fact_4237_Succ__Shift,axiom,
    ! [A: $tType,Kl: set(list(A)),K2: A,Kl2: list(A)] : bNF_Greatest_Succ(A,bNF_Greatest_Shift(A,Kl,K2),Kl2) = bNF_Greatest_Succ(A,Kl,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K2),Kl2)) ).

% Succ_Shift
tff(fact_4238_filter__filter,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool),Xs: list(A)] : filter2(A,P2,filter2(A,Q2,Xs)) = filter2(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qi(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2),Xs) ).

% filter_filter
tff(fact_4239_set__filter,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(list(A),set(A),set2(A),filter2(A,P2,Xs)) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_qj(fun(A,bool),fun(list(A),fun(A,bool)),P2),Xs)) ).

% set_filter
tff(fact_4240_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),K2: B,Xs: list(A)] : filter2(A,aa(B,fun(A,bool),aTP_Lamp_qk(fun(A,B),fun(B,fun(A,bool)),F3),K2),aa(list(A),list(A),linorder_sort_key(A,B,F3),Xs)) = filter2(A,aa(B,fun(A,bool),aTP_Lamp_qk(fun(A,B),fun(B,fun(A,bool)),F3),K2),Xs) ) ).

% sort_key_stable
tff(fact_4241_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool)] : pp(aa(set(list(A)),bool,member(list(A),Xs),shuffles(A,filter2(A,P2,Xs),filter2(A,aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2),Xs)))) ).

% partition_in_shuffles
tff(fact_4242_filter__is__subset,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),filter2(A,P2,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% filter_is_subset
tff(fact_4243_length__filter__le,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_filter_le
tff(fact_4244_sorted__filter_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),P2: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => sorted_wrt(A,ord_less_eq(A),filter2(A,P2,L)) ) ) ).

% sorted_filter'
tff(fact_4245_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fTrue,aTP_Lamp_ql(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(1)
tff(fact_4246_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fFalse,aTP_Lamp_qm(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(2)
tff(fact_4247_sum__length__filter__compl,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_4248_inter__set__filter,axiom,
    ! [A: $tType,A6: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),filter2(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6),Xs)) ).

% inter_set_filter
tff(fact_4249_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G3: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter2(A,aa(list(A),fun(A,bool),aTP_Lamp_qn(fun(list(A),A),fun(list(A),fun(A,bool)),G3),Xs),Xs)) ) ).

% sorted_same
tff(fact_4250_length__filter__less,axiom,
    ! [A: $tType,X: A,Xs: list(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P2,X))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% length_filter_less
tff(fact_4251_ShiftD,axiom,
    ! [A: $tType,Kl2: list(A),Kl: set(list(A)),K2: A] :
      ( pp(aa(set(list(A)),bool,member(list(A),Kl2),bNF_Greatest_Shift(A,Kl,K2)))
     => pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K2),Kl2)),Kl)) ) ).

% ShiftD
tff(fact_4252_filter__eq__snocD,axiom,
    ! [A: $tType,P2: fun(A,bool),L: list(A),L4: list(A),X: A] :
      ( ( filter2(A,P2,L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L)))
        & pp(aa(A,bool,P2,X)) ) ) ).

% filter_eq_snocD
tff(fact_4253_pick__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : pick(A,filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,bool)),fun(product_prod(code_natural,A),bool),product_case_prod(code_natural,A,bool),aTP_Lamp_qo(code_natural,fun(A,bool))),Xs)) = pick(A,Xs) ).

% pick_drop_zero
tff(fact_4254_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),Y),Xs)) ).

% set_minus_filter_out
tff(fact_4255_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( 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),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( filter2(A,aTP_Lamp_qq(list(A),fun(A,bool),Xs),Zs) = Ys2 ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_4256_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( 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),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( filter2(A,aTP_Lamp_qr(list(A),fun(A,bool),Xs),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_4257_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( 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),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( filter2(A,aTP_Lamp_qq(list(A),fun(A,bool),Ys2),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_4258_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( 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),Ys2)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
       => ( filter2(A,aTP_Lamp_qr(list(A),fun(A,bool),Ys2),Zs) = Ys2 ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_4259_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),filter2(A,P3,Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_qs(fun(A,bool),fun(list(A),fun(nat,bool)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_4260_Shift__def,axiom,
    ! [A: $tType,Kl: set(list(A)),K2: A] : bNF_Greatest_Shift(A,Kl,K2) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(A,fun(list(A),bool),aTP_Lamp_qt(set(list(A)),fun(A,fun(list(A),bool)),Kl),K2)) ).

% Shift_def
tff(fact_4261_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),filter2(A,P2,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,bool),set(A),collect(A),P2)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_4262_part__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Pivot: A,Xs: list(B)] : linorder_part(B,A,F3,Pivot,Xs) = aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),filter2(B,aa(A,fun(B,bool),aTP_Lamp_qu(fun(B,A),fun(A,fun(B,bool)),F3),Pivot),Xs)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),filter2(B,aa(A,fun(B,bool),aTP_Lamp_qv(fun(B,A),fun(A,fun(B,bool)),F3),Pivot),Xs)),filter2(B,aa(A,fun(B,bool),aTP_Lamp_qw(fun(B,A),fun(A,fun(B,bool)),F3),Pivot),Xs))) ) ).

% part_def
tff(fact_4263_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,bool)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P2) = aa(list(A),A,case_list(A,A,abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P2),aTP_Lamp_qx(A,fun(list(A),A))),filter2(A,P2,aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_4264_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))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_qy(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)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_qz(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_ra(list(B),bool),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_4265_merge__list_Opinduct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A0: list(list(A)),A1: list(list(A)),P2: fun(list(list(A)),fun(list(list(A)),bool))] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),A0),A1)))
         => ( ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A)))))
             => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,nil(list(A))),nil(list(A)))) )
           => ( ! [L3: list(A)] :
                  ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))))
                 => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))) )
             => ( ! [La: list(A),Acc22: list(list(A))] :
                    ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))))
                   => ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)))
                     => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))) ) )
               => ( ! [La: list(A),Acc22: list(list(A)),L3: list(A)] :
                      ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))))
                     => ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))))
                       => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))))) ) )
                 => ( ! [Acc22: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] :
                        ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)))))
                       => ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),Acc22)),Ls))
                         => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)))) ) )
                   => pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P2,A0),A1)) ) ) ) ) ) ) ) ).

% merge_list.pinduct
tff(fact_4266_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : aa(nat,nat,foldr(A,nat,aTP_Lamp_rb(A,fun(nat,nat)),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_4267_foldr__snd__zip,axiom,
    ! [B: $tType,A: $tType,C: $tType,Ys2: list(A),Xs: list(B),F3: fun(A,fun(C,C)),B2: C] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(B),nat,size_size(list(B)),Xs)))
     => ( aa(C,C,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_rc(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F3)),zip(B,A,Xs,Ys2)),B2) = aa(C,C,foldr(A,C,F3,Ys2),B2) ) ) ).

% foldr_snd_zip
tff(fact_4268_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A3: nat] : aa(nat,nat,foldr(A,nat,aTP_Lamp_rb(A,fun(nat,nat)),L),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_4269_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_rd(fun(B,A),fun(A,fun(B,fun(A,A))),F3),A3),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_4270_select__weight__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,bool)),fun(product_prod(code_natural,A),bool),product_case_prod(code_natural,A,bool),aTP_Lamp_qo(code_natural,fun(A,bool))),Xs)) = select_weight(A,Xs) ).

% select_weight_drop_zero
tff(fact_4271_merge__list_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(list(A)),Xa2: list(list(A)),Y: list(A)] :
          ( ( merge_list(A,X,Xa2) = Y )
         => ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),Xa2)))
           => ( ( ( X = nil(list(A)) )
               => ( ( Xa2 = nil(list(A)) )
                 => ( ( Y = nil(A) )
                   => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))) ) ) )
             => ( ( ( X = nil(list(A)) )
                 => ! [L3: list(A)] :
                      ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
                     => ( ( Y = L3 )
                       => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))))) ) ) )
               => ( ! [La: list(A),Acc22: list(list(A))] :
                      ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                     => ( ( Xa2 = nil(list(A)) )
                       => ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)) )
                         => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A))))) ) ) )
                 => ( ! [La: list(A),Acc22: list(list(A))] :
                        ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                       => ! [L3: list(A)] :
                            ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
                           => ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))) )
                             => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A)))))) ) ) )
                   => ~ ! [L1: list(A),L22: list(A),Ls: list(list(A))] :
                          ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)) )
                         => ( ( Y = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),X),Ls) )
                           => ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))))) ) ) ) ) ) ) ) ) ) ).

% merge_list.pelims
tff(fact_4272_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_re(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_rf(list(A),bool),Xs)) ).

% transpose_max_length
tff(fact_4273_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_rg(A,fun(A,bool),X),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aa(A,fun(A,bool),ord_less_eq(A),X),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_4274_Misc_Ofoldr__Cons,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),foldr(A,list(A),cons(A),Xs),nil(A)) = Xs ).

% Misc.foldr_Cons
tff(fact_4275_sort__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_pk(A,A)) = linorder_quicksort(A) ) ) ).

% sort_quicksort
tff(fact_4276_merge__list_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ).

% merge_list.simps(1)
tff(fact_4277_merge__list_Osimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A)),L: list(A)] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) ) ).

% merge_list.simps(4)
tff(fact_4278_merge__list_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A))] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) ) ).

% merge_list.simps(3)
tff(fact_4279_merge__list_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] : merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ).

% merge_list.simps(2)
tff(fact_4280_merge__list_Osimps_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Acc23: list(list(A)),L12: list(A),L23: list(A),Ls2: list(list(A))] : merge_list(A,Acc23,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L23)),Acc23),Ls2) ) ).

% merge_list.simps(5)
tff(fact_4281_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_4282_filter__conv__foldr,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : filter2(A,P2,Xs) = aa(list(A),list(A),foldr(A,list(A),aTP_Lamp_rh(fun(A,bool),fun(A,fun(list(A),list(A))),P2),Xs),nil(A)) ).

% filter_conv_foldr
tff(fact_4283_merge__list_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(list(A)),Xa2: list(list(A)),Y: list(A)] :
          ( ( merge_list(A,X,Xa2) = Y )
         => ( ( ( X = nil(list(A)) )
             => ( ( Xa2 = nil(list(A)) )
               => ( Y != nil(A) ) ) )
           => ( ( ( X = nil(list(A)) )
               => ! [L3: list(A)] :
                    ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
                   => ( Y != L3 ) ) )
             => ( ! [La: list(A),Acc22: list(list(A))] :
                    ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                   => ( ( Xa2 = nil(list(A)) )
                     => ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)) ) ) )
               => ( ! [La: list(A),Acc22: list(list(A))] :
                      ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22) )
                     => ! [L3: list(A)] :
                          ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),nil(list(A))) )
                         => ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L3),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))) ) ) )
                 => ~ ! [L1: list(A),L22: list(A),Ls: list(list(A))] :
                        ( ( Xa2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls)) )
                       => ( Y != merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L22)),X),Ls) ) ) ) ) ) ) ) ) ).

% merge_list.elims
tff(fact_4284_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_re(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_4285_merge__list_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A)))))
       => ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ) ).

% merge_list.psimps(1)
tff(fact_4286_merge__list_Opsimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A))))))
         => ( merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ) ) ).

% merge_list.psimps(2)
tff(fact_4287_merge__list_Opsimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A))] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),nil(list(A)))))
         => ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) ) ) ) ).

% merge_list.psimps(3)
tff(fact_4288_merge__list_Opsimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A)),L: list(A)] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A))))))
         => ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) ) ) ) ).

% merge_list.psimps(4)
tff(fact_4289_merge__list_Opsimps_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Acc23: list(list(A)),L12: list(A),L23: list(A),Ls2: list(list(A))] :
          ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2)))))
         => ( merge_list(A,Acc23,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L23),Ls2))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L23)),Acc23),Ls2) ) ) ) ).

% merge_list.psimps(5)
tff(fact_4290_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
         => ( ( ( X = nil(A) )
             => ( Y != nil(A) ) )
           => ~ ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( Y != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_rg(A,fun(A,bool),X3),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aa(A,fun(A,bool),ord_less_eq(A),X3),Xs2)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_4291_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),X) = Y )
         => ( pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),X))
           => ( ( ( X = nil(A) )
               => ( ( Y = nil(A) )
                 => ~ pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),nil(A))) ) )
             => ~ ! [X3: A,Xs2: list(A)] :
                    ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_rg(A,fun(A,bool),X3),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aa(A,fun(A,bool),ord_less_eq(A),X3),Xs2)))) )
                     => ~ pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_4292_distinct__concat_H,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),filter2(list(A),aTP_Lamp_rf(list(A),bool),Xs))
     => ( ! [Ys5: list(A)] :
            ( pp(aa(set(list(A)),bool,member(list(A),Ys5),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys5) )
       => ( ! [Ys5: list(A),Zs2: list(A)] :
              ( pp(aa(set(list(A)),bool,member(list(A),Ys5),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
             => ( pp(aa(set(list(A)),bool,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_4293_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),bool),aTP_Lamp_ri(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_4294_execute__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,X: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_new(A,N,X)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rj(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,replicate(A,N,X),H))) ) ).

% execute_new
tff(fact_4295_nth__replicate,axiom,
    ! [A: $tType,I2: nat,N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
     => ( aa(nat,A,nth(A,replicate(A,N,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_4296_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),Xs) )
         => ? [N4: nat,Zs2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N4))
              & ( concat(A,replicate(list(A),N4,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_4297_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: nat,X: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,N,X)) ) ).

% sorted_replicate
tff(fact_4298_replicate__add,axiom,
    ! [A: $tType,N: nat,M: nat,X: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X)),replicate(A,M,X)) ).

% replicate_add
tff(fact_4299_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs)),X) = filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs) ).

% replicate_length_filter
tff(fact_4300_new__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,X: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,N,X),aa(A,fun(array(A),assn),aTP_Lamp_rk(nat,fun(A,fun(array(A),assn)),N),X)) ) ).

% new_rule
tff(fact_4301_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs: list(list(A))] : concat(A,filter2(list(A),aTP_Lamp_rf(list(A),bool),Xs)) = concat(A,Xs) ).

% concat_filter_neq_Nil
tff(fact_4302_foldl__foldl__conv__concat,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,A)),A3: A,Xs: list(list(B))] : aa(list(list(B)),A,aa(A,fun(list(list(B)),A),foldl(A,list(B),foldl(A,B,F3)),A3),Xs) = aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),A3),concat(B,Xs)) ).

% foldl_foldl_conv_concat
tff(fact_4303_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% replicate_Suc_conv_snoc
tff(fact_4304_array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,X: A] : array_new(A,N,X) = array_make(A,N,aTP_Lamp_rl(A,fun(nat,A),X)) ) ).

% array_make
tff(fact_4305_effect__newI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H5: heap_ext(product_unit),N: nat,X: A,H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H5) = array_alloc(A,replicate(A,N,X),H) )
         => heap_Time_effect(array(A),array_new(A,N,X),H,H5,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_newI
tff(fact_4306_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = replicate(A,N,Y) )
    <=> ( ( X = Y )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_4307_merge__list__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ls2: list(list(A)),As: list(list(A))] :
          ( ! [L3: list(A)] :
              ( pp(aa(set(list(A)),bool,member(list(A),L3),aa(list(list(A)),set(list(A)),set2(list(A)),Ls2)))
             => ( distinct(A,L3)
                & sorted_wrt(A,ord_less_eq(A),L3) ) )
         => ( ! [L3: list(A)] :
                ( pp(aa(set(list(A)),bool,member(list(A),L3),aa(list(list(A)),set(list(A)),set2(list(A)),As)))
               => ( distinct(A,L3)
                  & sorted_wrt(A,ord_less_eq(A),L3) ) )
           => ( distinct(A,merge_list(A,As,Ls2))
              & sorted_wrt(A,ord_less_eq(A),merge_list(A,As,Ls2))
              & ( aa(list(A),set(A),set2(A),merge_list(A,As,Ls2)) = aa(list(A),set(A),set2(A),concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),As),Ls2))) ) ) ) ) ) ).

% merge_list_correct
tff(fact_4308_new__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,X: A] : array_new(A,N,X) = heap_Time_heap(array(A),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rm(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),X)) ) ).

% new_def
tff(fact_4309_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),listrel1(A,R3)))
    <=> ? [Y4: A,N3: nat] :
          ( pp(aa(set(product_prod(A,A)),bool,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),N3)),Y4)),R3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( Ys2 = list_update(A,Xs,N3,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_4310_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_pk(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_4311_mset__mergesort__by__rel__split,axiom,
    ! [A: $tType,Xs1: list(A),Xs22: list(A),Xs: list(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs)))),mset(A,aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs)))) = aa(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)),mset(A,Xs)),mset(A,Xs1))),mset(A,Xs22)) ).

% mset_mergesort_by_rel_split
tff(fact_4312_nth__transpose,axiom,
    ! [A: $tType,I2: nat,Xs: list(list(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_rn(nat,fun(list(A),A),I2)),filter2(list(A),aTP_Lamp_ro(nat,fun(list(A),bool),I2),Xs)) ) ) ).

% nth_transpose
tff(fact_4313_map__ident,axiom,
    ! [A: $tType,X5: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_cc(A,A)),X5) = X5 ).

% map_ident
tff(fact_4314_mergesort__by__rel__permutes,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A)] : mset(A,aa(list(A),list(A),mergesort_by_rel(A,R2),Xs)) = mset(A,Xs) ).

% mergesort_by_rel_permutes
tff(fact_4315_mset__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : mset(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys2)) ).

% mset_append
tff(fact_4316_mset__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] : mset(A,merges9089515139780605204_merge(A,R2,Xs,Ys2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys2)) ).

% mset_mergesort_by_rel_merge
tff(fact_4317_Multiset_Omset__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : mset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X),Xs)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),mset(A,Xs)) ) ).

% Multiset.mset_insort
tff(fact_4318_nth__map,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),F3: fun(A,B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F3),Xs)),N) = aa(A,B,F3,aa(nat,A,nth(A,Xs),N)) ) ) ).

% nth_map
tff(fact_4319_sorted__list__of__multiset__mset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : linord6283353356039996273ltiset(A,mset(A,Xs)) = aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),Xs) ) ).

% sorted_list_of_multiset_mset
tff(fact_4320_Cons__listrel1__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),listrel1(A,R3)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
          & ( Xs = Ys2 ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4321_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_rp(fun(B,A),fun(B,list(A)),F3)),Xs)) = aa(list(B),list(A),map(B,A,F3),Xs) ).

% concat_map_singleton
tff(fact_4322_distinct__mapI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => distinct(B,L) ) ).

% distinct_mapI
tff(fact_4323_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(product_prod(B,C),A),Xs: list(B),G3: fun(D,C),Ys2: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G3),Ys2))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_rq(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F3),G3))),zip(B,D,Xs,Ys2)) ).

% map_zip_map2
tff(fact_4324_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(product_prod(B,C),A),G3: fun(D,B),Xs: list(D),Ys2: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,aa(list(D),list(B),map(D,B,G3),Xs),Ys2)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_rr(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F3),G3))),zip(D,C,Xs,Ys2)) ).

% map_zip_map
tff(fact_4325_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F3: fun(D,B),Xs: list(D),G3: fun(D,C)] : aa(list(product_prod(B,C)),list(A),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,aa(list(D),list(B),map(D,B,F3),Xs),aa(list(D),list(C),map(D,C,G3),Xs))) = aa(list(D),list(A),map(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_rs(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F3),G3)),Xs) ).

% map2_map_map
tff(fact_4326_Misc_Omap__eq__append__conv,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls2: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
    <=> ? [L2: list(B),L5: list(B)] :
          ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L5) )
          & ( aa(list(B),list(A),map(B,A,F3),L2) = Fl )
          & ( aa(list(B),list(A),map(B,A,F3),L5) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_4327_Misc_Oappend__eq__map__conv,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F3: fun(B,A),Ls2: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F3),Ls2) )
    <=> ? [L2: list(B),L5: list(B)] :
          ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L5) )
          & ( aa(list(B),list(A),map(B,A,F3),L2) = Fl )
          & ( aa(list(B),list(A),map(B,A,F3),L5) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_4328_map__eq__appendE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls2: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
     => ~ ! [L3: list(B),L6: list(B)] :
            ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L6) )
           => ( ( aa(list(B),list(A),map(B,A,F3),L3) = Fl )
             => ( aa(list(B),list(A),map(B,A,F3),L6) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_4329_append__eq__mapE,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F3: fun(B,A),Ls2: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F3),Ls2) )
     => ~ ! [L3: list(B),L6: list(B)] :
            ( ( Ls2 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L6) )
           => ( ( aa(list(B),list(A),map(B,A,F3),L3) = Fl )
             => ( aa(list(B),list(A),map(B,A,F3),L6) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_4330_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B),L4: list(B),I2: nat] :
      ( ( aa(list(B),list(A),map(B,A,F3),L) = aa(list(B),list(A),map(B,A,F3),L4) )
     => ( aa(B,A,F3,aa(nat,B,nth(B,L),I2)) = aa(B,A,F3,aa(nat,B,nth(B,L4),I2)) ) ) ).

% map_eq_nth_eq
tff(fact_4331_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,bool)),F3: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R2,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_rt(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R2),F3),Xs) ) ).

% sorted_wrt_map
tff(fact_4332_listrel1__mono,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel1(A,S2))) ) ).

% listrel1_mono
tff(fact_4333_list_Omap__ident,axiom,
    ! [A: $tType,T5: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_cc(A,A)),T5) = T5 ).

% list.map_ident
tff(fact_4334_map__eq__consE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls2: list(B),Fa: A,Fl: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
     => ~ ! [A5: B,L3: list(B)] :
            ( ( Ls2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A5),L3) )
           => ( ( aa(B,A,F3,A5) = Fa )
             => ( aa(list(B),list(A),map(B,A,F3),L3) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_4335_map__consI_I1_J,axiom,
    ! [A: $tType,B: $tType,W2: list(A),F3: fun(B,A),Ww: list(B),A3: B] :
      ( ( W2 = aa(list(B),list(A),map(B,A,F3),Ww) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F3,A3)),W2) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww)) ) ) ).

% map_consI(1)
tff(fact_4336_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G3: fun(A,fun(B,A)),A3: A,F3: fun(C,B),Xs: list(C)] : aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,G3),A3),aa(list(C),list(B),map(C,B,F3),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_ru(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G3),F3)),A3),Xs) ).

% foldl_map
tff(fact_4337_map__prod__fun__zip,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),Xs: list(C),Ys2: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_rv(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),zip(C,D,Xs,Ys2)) = zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),aa(list(D),list(B),map(D,B,G3),Ys2)) ).

% map_prod_fun_zip
tff(fact_4338_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F3: fun(C,B),Ys2: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F3),Ys2)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),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_rw(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),zip(A,C,Xs,Ys2)) ).

% zip_map2
tff(fact_4339_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xs: list(C),Ys2: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),Ys2) = aa(list(product_prod(C,B)),list(product_prod(A,B)),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_rx(fun(C,A),fun(C,fun(B,product_prod(A,B))),F3))),zip(C,B,Xs,Ys2)) ).

% zip_map1
tff(fact_4340_mset__eq__finite,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_ry(list(A),fun(list(A),bool),Xs)))) ).

% mset_eq_finite
tff(fact_4341_map__consI_I2_J,axiom,
    ! [B: $tType,A: $tType,W2: list(A),L: list(A),F3: fun(B,A),Ww: list(B),A3: B] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F3),Ww)),L) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F3,A3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),L)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww))),L) ) ) ).

% map_consI(2)
tff(fact_4342_distinct__map__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B),X: B,Y: B] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => ( ( aa(B,A,F3,X) = aa(B,A,F3,Y) )
       => ( pp(aa(set(B),bool,member(B,X),aa(list(B),set(B),set2(B),L)))
         => ( pp(aa(set(B),bool,member(B,Y),aa(list(B),set(B),set2(B),L)))
           => ( X = Y ) ) ) ) ) ).

% distinct_map_eq
tff(fact_4343_mset__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys2: list(A)] :
      ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys2)))
     => ( mset(A,Zs) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys2)) ) ) ).

% mset_shuffles
tff(fact_4344_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
        <=> sorted_wrt(B,aTP_Lamp_rz(fun(B,A),fun(B,fun(B,bool)),F3),Xs) ) ) ).

% sorted_map
tff(fact_4345_mset__eq__length__filter,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Z4: A] :
      ( ( mset(A,Xs) = mset(A,Ys2) )
     => ( aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,bool),fequal(A),Z4),Xs)) = aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,bool),fequal(A),Z4),Ys2)) ) ) ).

% mset_eq_length_filter
tff(fact_4346_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K2: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_or(A,fun(B,A)),K2)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K2) ).

% map_replicate_const
tff(fact_4347_properties__for__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Ys2: list(B),Xs: list(B),F3: fun(B,A)] :
          ( ( mset(B,Ys2) = mset(B,Xs) )
         => ( ! [K: B] :
                ( pp(aa(set(B),bool,member(B,K),aa(list(B),set(B),set2(B),Ys2)))
               => ( filter2(B,aa(B,fun(B,bool),aTP_Lamp_sa(fun(B,A),fun(B,fun(B,bool)),F3),K),Ys2) = filter2(B,aa(B,fun(B,bool),aTP_Lamp_sa(fun(B,A),fun(B,fun(B,bool)),F3),K),Xs) ) )
           => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Ys2))
             => ( aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = Ys2 ) ) ) ) ) ).

% properties_for_sort_key
tff(fact_4348_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),filter2(B,P2,Xs))) ) ) ).

% sorted_filter
tff(fact_4349_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ).

% sorted_insort_key
tff(fact_4350_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs))) ) ).

% sorted_sort_key
tff(fact_4351_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),G3: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),filter2(B,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_sb(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F3),G3),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_4352_properties__for__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ys2: list(A),Xs: list(A)] :
          ( ( mset(A,Ys2) = mset(A,Xs) )
         => ( sorted_wrt(A,ord_less_eq(A),Ys2)
           => ( aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),Xs) = Ys2 ) ) ) ) ).

% properties_for_sort
tff(fact_4353_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,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),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),listrel1(A,R3)))
     => ( ! [X3: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2) )
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y)),R3)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,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)),Zs2),Ys2)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1E2
tff(fact_4354_Cons__listrel1E1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys2)),listrel1(A,R3)))
     => ( ! [Y3: A] :
            ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs) )
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R3)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,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),Zs2)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1E1
tff(fact_4355_listrel1I1,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))),listrel1(A,R3))) ) ).

% listrel1I1
tff(fact_4356_transpose_Oelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( ( ( X = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_sc(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_sd(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_4357_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_sc(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_sd(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_4358_distinct__idx,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),L: list(B),I2: nat,J2: nat] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),L)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(B),nat,size_size(list(B)),L)))
         => ( ( aa(B,A,F3,aa(nat,B,nth(B,L),I2)) = aa(B,A,F3,aa(nat,B,nth(B,L),J2)) )
           => ( I2 = J2 ) ) ) ) ) ).

% distinct_idx
tff(fact_4359_mset__swap,axiom,
    ! [A: $tType,I2: nat,Ls2: list(A),J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ls2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Ls2)))
       => ( mset(A,list_update(A,list_update(A,Ls2,J2,aa(nat,A,nth(A,Ls2),I2)),I2,aa(nat,A,nth(A,Ls2),J2))) = mset(A,Ls2) ) ) ) ).

% mset_swap
tff(fact_4360_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I2: nat,L: list(A),F3: fun(A,B),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
       => ( aa(A,B,F3,aa(nat,A,nth(A,L),I2)) = aa(A,B,F3,X) ) )
     => ( aa(list(A),list(B),map(A,B,F3),list_update(A,L,I2,X)) = aa(list(A),list(B),map(A,B,F3),L) ) ) ).

% map_upd_eq
tff(fact_4361_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,bool),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => ( pp(aa(B,bool,P2,X))
           => ( filter2(B,P2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),filter2(B,P2,Xs)) ) ) ) ) ).

% filter_insort
tff(fact_4362_listrel1I,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys2: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
       => ( ( Ys2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) )
         => pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),listrel1(A,R3))) ) ) ) ).

% listrel1I
tff(fact_4363_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),listrel1(A,R3)))
     => ~ ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
           => ! [Us2: list(A),Vs2: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Vs2)) )
               => ( Ys2 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_4364_map__by__foldl,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),L: list(A)] : aa(list(A),list(B),aa(list(B),fun(list(A),list(B)),foldl(list(B),A,aTP_Lamp_se(fun(A,B),fun(list(B),fun(A,list(B))),F3)),nil(B)),L) = aa(list(A),list(B),map(A,B,F3),L) ).

% map_by_foldl
tff(fact_4365_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys2: list(A),Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),listrel1(A,R3)))
    <=> ( ( pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),listrel1(A,R3)))
          & ( X = Y ) )
        | ( ( Xs = Ys2 )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_4366_mset__update,axiom,
    ! [A: $tType,I2: nat,Ls2: list(A),V2: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ls2)))
     => ( mset(A,list_update(A,Ls2,I2,V2)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),V2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),mset(A,Ls2)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),aa(nat,A,nth(A,Ls2),I2)),zero_zero(multiset(A))))) ) ) ).

% mset_update
tff(fact_4367_listrel1p__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] :
      ( listrel1p(A,R3,Xs,Ys2)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),listrel1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% listrel1p_def
tff(fact_4368_transpose_Opelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),X))
       => ( ( ( X = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A)))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
           => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_sc(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_sd(A,fun(list(A),list(list(A)))))),Xss2))))) )
                   => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_4369_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_sc(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_sd(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_4370_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat,J2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_ro(nat,fun(list(A),bool),I2),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2)),J2) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J2)),I2) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_4371_sorted__wrt__rev__linord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,aTP_Lamp_mo(A,fun(A,bool)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_4372_sorted__wrt__map__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_sf(product_prod(A,B),fun(product_prod(A,B),bool)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L)) ) ) ).

% sorted_wrt_map_linord
tff(fact_4373_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K2: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_oz(B,fun(A,product_prod(A,B))),K2)),L)) = L ).

% map_fst_mk_snd
tff(fact_4374_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K2: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(B,A)),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2)),L)) = L ).

% map_snd_mk_fst
tff(fact_4375_sorted__wrt__map__rev__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_sg(product_prod(A,B),fun(product_prod(A,B),bool)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))) ) ) ).

% sorted_wrt_map_rev_linord
tff(fact_4376_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K2: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K2) ).

% map_fst_mk_fst
tff(fact_4377_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K2: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(B),list(product_prod(B,A)),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_pc(A,fun(B,product_prod(B,A))),K2)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K2) ).

% map_snd_mk_snd
tff(fact_4378_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : product(A,B,Xs,Ys2) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_sh(list(B),fun(A,list(product_prod(A,B))),Ys2)),Xs)) ).

% product_concat_map
tff(fact_4379_sorted__wrt__rev,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P2,rev(A,Xs))
    <=> sorted_wrt(A,aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P2),Xs) ) ).

% sorted_wrt_rev
tff(fact_4380_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P2: fun(nat,bool),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_sj(fun(nat,bool),fun(product_prod(A,nat),bool),P2),zip(A,nat,Xs,Is))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_sk(fun(nat,bool),fun(product_prod(A,nat),bool),P2),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_4381_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_ox(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_4382_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B),A3: A] : aa(A,A,foldr(B,A,F3,Xs),A3) = aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,aTP_Lamp_sl(fun(B,fun(A,A)),fun(A,fun(B,A)),F3)),A3),rev(B,Xs)) ).

% foldr_conv_foldl
tff(fact_4383_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),A3),Xs) = aa(A,A,foldr(B,A,aTP_Lamp_sm(fun(A,fun(B,A)),fun(B,fun(A,A)),F3),rev(B,Xs)),A3) ).

% foldl_conv_foldr
tff(fact_4384_distinct__map__fst__filterI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),P2: fun(product_prod(A,B),bool)] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),filter2(product_prod(A,B),P2,Xs))) ) ).

% distinct_map_fst_filterI
tff(fact_4385_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys2: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys2,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),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_sn(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys2),Zs)) ).

% zip_assoc
tff(fact_4386_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : zip(A,B,Xs,Ys2) = aa(list(product_prod(B,A)),list(product_prod(A,B)),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_oz(B,fun(A,product_prod(A,B))))),zip(B,A,Ys2,Xs)) ).

% zip_commute
tff(fact_4387_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_sp(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_4388_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K2: A,V1: B,V22: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V1)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V22)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_4389_distinct__map__fstD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),X: A,Y: B,Z4: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( pp(aa(set(product_prod(A,B)),bool,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)),Xs)))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z4)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
         => ( Y = Z4 ) ) ) ) ).

% distinct_map_fstD
tff(fact_4390_map__zip2,axiom,
    ! [A: $tType,B: $tType,K2: A,L: list(B)] : aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2)),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K2),L) ).

% map_zip2
tff(fact_4391_map__zip1,axiom,
    ! [A: $tType,B: $tType,K2: B,L: list(A)] : aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_oz(B,fun(A,product_prod(A,B))),K2)),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K2)) ).

% map_zip1
tff(fact_4392_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_4393_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys2: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),Ys2)),product(A,B,Xs,Ys2)) ).

% product.simps(2)
tff(fact_4394_rev__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rev(A,Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,N))) ) ) ).

% rev_nth
tff(fact_4395_rev__update,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( rev(A,list_update(A,Xs,K2,Y)) = list_update(A,rev(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K2)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_4396_sorted__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,N,Xs))) ).

% sorted_enumerate
tff(fact_4397_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I))),aa(nat,A,nth(A,Xs),I))) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_4398_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I))) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_4399_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J2)),aa(nat,A,nth(A,Xs),I2))) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_4400_mergesort__remdups__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),mergesort_remdups(A),Xs) = merge_list(A,nil(list(A)),aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_sq(A,list(A))),Xs)) ) ).

% mergesort_remdups_def
tff(fact_4401_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys2: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys2,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),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_ss(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys2,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_4402_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( ( ( Xs = nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y) = Y ) )
            & ( ( Xs != nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),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_4403_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P2: fun(list(list(A)),bool)] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),A0))
     => ( ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A))))
         => pp(aa(list(list(A)),bool,P2,nil(list(A)))) )
       => ( ! [Xss2: list(list(A))] :
              ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)))
             => ( pp(aa(list(list(A)),bool,P2,Xss2))
               => pp(aa(list(list(A)),bool,P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
         => ( ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2)))
               => ( pp(aa(list(list(A)),bool,P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_sd(A,fun(list(A),list(list(A)))))),Xss2)))))
                 => pp(aa(list(list(A)),bool,P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) )
           => pp(aa(list(list(A)),bool,P2,A0)) ) ) ) ) ).

% transpose.pinduct
tff(fact_4404_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( ( ( Xs = nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
        & ( ( Xs != nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).

% length_transpose_sorted
tff(fact_4405_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_ro(nat,fun(list(A),bool),I2),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) ) ) ) ).

% transpose_column_length
tff(fact_4406_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_rn(nat,fun(list(A),A),I2)),filter2(list(A),aTP_Lamp_ro(nat,fun(list(A),bool),I2),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I2) ) ) ) ).

% transpose_column
tff(fact_4407_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys2)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_sh(list(B),fun(A,list(product_prod(A,B))),Ys2)),Xs))) ).

% product_code
tff(fact_4408_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),Xss)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_st(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_4409_execute__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,F3: fun(nat,A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_make(A,N,F3)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rj(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),N)),H))) ) ).

% execute_make
tff(fact_4410_map__filter__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F3,Xs) = aa(list(A),list(B),map(A,B,aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),F3)),filter2(A,aTP_Lamp_su(fun(A,option(B)),fun(A,bool),F3),Xs)) ).

% map_filter_def
tff(fact_4411_sort__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),linorder_sort_key(nat,nat,aTP_Lamp_fn(nat,nat)),upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_4412_upt__0__eq__Nil__conv,axiom,
    ! [J2: nat] :
      ( ( upt(zero_zero(nat),J2) = nil(nat) )
    <=> ( J2 = zero_zero(nat) ) ) ).

% upt_0_eq_Nil_conv
tff(fact_4413_upt__merge,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),K2)) )
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J2)),upt(J2,K2)) = upt(I2,K2) ) ) ).

% upt_merge
tff(fact_4414_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( upt(I2,J2) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_4415_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( upt(I2,J2) = nil(nat) )
    <=> ( ( J2 = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2)) ) ) ).

% upt_eq_Nil_conv
tff(fact_4416_nth__upt,axiom,
    ! [I2: nat,K2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J2))
     => ( aa(nat,nat,nth(nat,upt(I2,J2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2) ) ) ).

% nth_upt
tff(fact_4417_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_4418_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_4419_upt__eq__append__conv,axiom,
    ! [I2: nat,J2: nat,Xs: list(nat),Ys2: list(nat)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( ( upt(I2,J2) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xs),Ys2) )
      <=> ? [K3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),J2))
            & ( upt(I2,K3) = Xs )
            & ( upt(K3,J2) = Ys2 ) ) ) ) ).

% upt_eq_append_conv
tff(fact_4420_map__add__upt_H,axiom,
    ! [Ofs: nat,A3: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_sv(nat,fun(nat,nat),Ofs)),upt(A3,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_4421_upt__append,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I2)),upt(I2,J2)) = upt(zero_zero(nat),J2) ) ) ).

% upt_append
tff(fact_4422_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J2)),upt(J2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ) ).

% upt_add_eq_append
tff(fact_4423_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M,N)) ).

% sorted_wrt_upt
tff(fact_4424_map__add__upt,axiom,
    ! [N: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_sv(nat,fun(nat,nat),N)),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_4425_sorted__upt,axiom,
    ! [M: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,N)) ).

% sorted_upt
tff(fact_4426_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I2: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_sw(A,fun(nat,A),X)),upt(zero_zero(nat),I2)) = replicate(A,I2,X) ).

% map_replicate_trivial
tff(fact_4427_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,F3: fun(nat,A),M: nat] : enumerate(A,N,aa(list(nat),list(A),map(nat,A,F3),upt(N,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_sx(fun(nat,A),fun(nat,product_prod(nat,A)),F3)),upt(N,M)) ).

% enumerate_map_upt
tff(fact_4428_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( upt(I2,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J2)) ) ) ).

% upt_conv_Cons
tff(fact_4429_upt__rec,axiom,
    ! [I2: nat,J2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
       => ( upt(I2,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
       => ( upt(I2,J2) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_4430_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => ( upt(I2,aa(nat,nat,suc,J2)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J2)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => ( upt(I2,aa(nat,nat,suc,J2)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_4431_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( upt(I2,aa(nat,nat,suc,J2)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J2)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_4432_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_option(list(A),A,map_filter(B,A,F3,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_sy(fun(B,option(A)),fun(list(B),fun(A,list(A))),F3),Xs),aa(B,option(A),F3,X)) ).

% map_filter_simps(1)
tff(fact_4433_enumerate__eq__zip,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = zip(nat,A,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_4434_map__upt__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),N: nat] : aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F3,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_sz(fun(nat,A),fun(nat,A),F3)),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_4435_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),U4))
     => ( ! [I3: nat] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),I3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),U4)) )
           => ~ pp(aa(nat,bool,P2,I3)) )
       => ( filter2(nat,P2,upt(zero_zero(nat),U)) = filter2(nat,P2,upt(zero_zero(nat),U4)) ) ) ) ).

% upt_filter_extend
tff(fact_4436_map__decr__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_gt(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_4437_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(nat),list(A),map(nat,A,nth(A,Xs)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_4438_nth__map__upt,axiom,
    ! [A: $tType,I2: nat,N: nat,M: nat,F3: fun(nat,A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F3),upt(M,N))),I2) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) ) ).

% nth_map_upt
tff(fact_4439_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list(nat),I2: nat,Is2: list(nat)] :
      ( ( upt(L,H) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),Is2)) )
    <=> ( ( Is1 = upt(L,I2) )
        & ( Is2 = upt(aa(nat,nat,suc,I2),H) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),I2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),H)) ) ) ).

% upt_eq_lel_conv
tff(fact_4440_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J2: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I2,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
        & ( I2 = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)),J2) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_4441_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M: nat,A3: A] : enumerate(A,N,replicate(A,M,A3)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_ta(A,fun(nat,product_prod(nat,A)),A3)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% enumerate_replicate_eq
tff(fact_4442_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M: nat,F3: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I3)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F3),upt(M,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_4443_filter__upt__last,axiom,
    ! [A: $tType,P2: fun(A,bool),L: list(A),Js: list(nat),J2: nat,I2: nat] :
      ( ( filter2(nat,aa(list(A),fun(nat,bool),aTP_Lamp_tb(fun(A,bool),fun(list(A),fun(nat,bool)),P2),L),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),I2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
         => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,L),I2))) ) ) ) ).

% filter_upt_last
tff(fact_4444_nths__shift__lemma,axiom,
    ! [A: $tType,A6: set(nat),Xs: list(A),I2: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_tc(set(nat),fun(product_prod(A,nat),bool),A6),zip(A,nat,Xs,upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_td(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A6),I2),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_4445_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,F3: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,N,F3),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_te(nat,fun(fun(nat,A),fun(array(A),assn)),N),F3)) ) ).

% make_rule
tff(fact_4446_effect__makeI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H5: heap_ext(product_unit),F3: fun(nat,A),N: nat,H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H5) = array_alloc(A,aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),N)),H) )
         => heap_Time_effect(array(A),array_make(A,N,F3),H,H5,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_makeI
tff(fact_4447_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I3)) = N ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_tg(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_4448_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P2: fun(B,bool),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),filter2(B,P2,Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_th(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F3),P2),Xs) ).

% map_filter_map_filter
tff(fact_4449_Array__Time_Omake__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,F3: fun(nat,A)] : array_make(A,N,F3) = heap_Time_heap(array(A),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ti(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),F3)) ) ).

% Array_Time.make_def
tff(fact_4450_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A6: set(nat)] : nths(A,Xs,A6) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_tc(set(nat),fun(product_prod(A,nat),bool),A6),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_4451_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F3: fun(nat,B),Ns: list(nat)] :
          ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F3,X3)),aa(nat,B,F3,Y3))) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),groups7311177749621191930dd_sum(nat,B,F3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(nat),list(B),map(nat,B,F3),Ns)))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4452_remove__rev__alt__def,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),remove_rev(A,X),Xs) = filter2(A,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),X),rev(A,Xs)) ).

% remove_rev_alt_def
tff(fact_4453_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_sd(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),filter2(list(A),aTP_Lamp_rf(list(A),bool),Xss)) ).

% transpose_aux_filter_tail
tff(fact_4454_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ).

% sum_list.Cons
tff(fact_4455_sum__list__append,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A),Ys2: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),aa(list(A),A,groups8242544230860333062m_list(A),Ys2)) ) ).

% sum_list_append
tff(fact_4456_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aTP_Lamp_tj(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_4457_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M,N)) = groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fn(nat,nat),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% sum_list_upt
tff(fact_4458_in__set__tlD,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))))
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_tlD
tff(fact_4459_tl__obtain__elem,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
       => ~ ! [E2: A] : Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) ) ) ).

% tl_obtain_elem
tff(fact_4460_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_4461_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% member_le_sum_list
tff(fact_4462_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fz(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F3),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G3),Xs))) ) ).

% sum_list_addf
tff(fact_4463_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),C3: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_gc(fun(B,A),fun(A,fun(B,A)),F3),C3)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F3),Xs))),C3) ) ).

% sum_list_mult_const
tff(fact_4464_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C3: A,F3: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gd(A,fun(fun(B,A),fun(B,A)),C3),F3)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F3),Xs))) ) ).

% sum_list_const_mult
tff(fact_4465_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ge(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F3),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G3),Xs))) ) ).

% sum_list_subtractf
tff(fact_4466_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I5))),aa(list(A),set(A),set2(A),Xs))) ).

% set_nths_subset
tff(fact_4467_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(set(nat),bool,member(nat,I3),I5)) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_4468_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_4469_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_tk(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_4470_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys2),aTP_Lamp_tl(list(A),fun(A,fun(list(A),list(A))),Ys2)),Xs) ).

% tl_append
tff(fact_4471_tl__subset,axiom,
    ! [A: $tType,Xs: list(A),A6: set(A)] :
      ( ( Xs != nil(A) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))),A6)) ) ) ).

% tl_subset
tff(fact_4472_Misc_Onth__tl,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% Misc.nth_tl
tff(fact_4473_list__take__induct__tl2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),P2: fun(B,fun(A,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( ! [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(B,fun(A,bool),P2,aa(nat,B,nth(B,Ys2),N4)),aa(nat,A,nth(A,Xs),N4))) )
       => ! [N8: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N8),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
           => pp(aa(A,bool,aa(B,fun(A,bool),P2,aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Ys2)),N8)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N8))) ) ) ) ).

% list_take_induct_tl2
tff(fact_4474_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A))) ) ) ).

% sum_list_nonpos
tff(fact_4475_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_4476_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_4477_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),map(A,A,abs_abs(A)),Xs)))) ) ).

% sum_list_abs
tff(fact_4478_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(A,A,foldr(A,A,plus_plus(A),Xs),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_4479_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F3),filter2(A,P2,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F3),Xs)))) ).

% sum_list_filter_le_nat
tff(fact_4480_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))) )
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F3),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,G3),Xs)))) ) ) ).

% sum_list_mono
tff(fact_4481_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F3: fun(B,A),P2: fun(B,bool),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F3),filter2(B,P2,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_tm(fun(B,A),fun(fun(B,bool),fun(B,A)),F3),P2)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_4482_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = groups7311177749621191930dd_sum(A,A,aTP_Lamp_tn(A,A),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_4483_List_Onth__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% List.nth_tl
tff(fact_4484_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F3),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,G3),Xs)))) ) ) ) ).

% sum_list_strict_mono
tff(fact_4485_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K2: nat,Ns: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Ns)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Ns),K2)),aa(list(A),A,groups8242544230860333062m_list(A),Ns))) ) ) ).

% elem_le_sum_list
tff(fact_4486_nths__append,axiom,
    ! [A: $tType,L: list(A),L4: list(A),A6: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L4),A6) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A6)),nths(A,L4,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_to(list(A),fun(set(nat),fun(nat,bool)),L),A6)))) ).

% nths_append
tff(fact_4487_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S2: set(nat)] :
      ( distinct(A,Xs)
     => ( filter2(A,aa(set(nat),fun(A,bool),aTP_Lamp_tp(list(A),fun(set(nat),fun(A,bool)),Xs),S2),Xs) = nths(A,Xs,S2) ) ) ).

% filter_in_nths
tff(fact_4488_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_tq(list(A),fun(set(nat),fun(nat,bool)),Xs),I5))) ).

% length_nths
tff(fact_4489_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R3: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aTP_Lamp_tr(B,fun(C,B),R3)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R3) ) ).

% sum_list_triv
tff(fact_4490_sum__list__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,aTP_Lamp_mq(fun(A,nat),fun(A,nat),F3)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_4491_filter__eq__nths,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : filter2(A,P2,Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_qs(fun(A,bool),fun(list(A),fun(nat,bool)),P2),Xs))) ).

% filter_eq_nths
tff(fact_4492_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = groups7311177749621191930dd_sum(nat,B,nth(B,Xs),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_4493_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_ts(nat,fun(nat,fun(list(nat),bool)),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),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_tt(nat,fun(nat,fun(list(nat),bool)),M),N7)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_tu(nat,fun(nat,fun(list(nat),bool)),M),N7)))) ) ) ).

% card_length_sum_list_rec
tff(fact_4494_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_ts(nat,fun(nat,fun(list(nat),bool)),M),N7))) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(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_4495_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F3),Xs)) = groups7311177749621191930dd_sum(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_tv(fun(A,nat),fun(list(A),fun(A,nat)),F3),Xs),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_4496_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K2: nat,Xs: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),list_update(A,Xs,K2,X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),X)),aa(nat,A,nth(A,Xs),K2)) ) ) ) ).

% sum_list_update
tff(fact_4497_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A6: set(nat)] : nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),A6) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),if(list(A),aa(set(nat),bool,member(nat,zero_zero(nat)),A6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),nil(A))),nths(A,L,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_tw(set(nat),fun(nat,bool),A6)))) ).

% nths_Cons
tff(fact_4498_pick__member,axiom,
    ! [A: $tType,I2: code_natural,Xs: list(product_prod(code_natural,A))] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I2),aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))))
     => pp(aa(set(A),bool,member(A,aa(code_natural,A,pick(A,Xs),I2)),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),map(product_prod(code_natural,A),A,product_snd(code_natural,A)),Xs)))) ) ).

% pick_member
tff(fact_4499_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F3),Xs)) = groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_tx(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F3),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_4500_select__weight__def,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))),aTP_Lamp_ty(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).

% select_weight_def
tff(fact_4501_select__weight__member,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A)),S2: product_prod(code_natural,code_natural)] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))))
     => pp(aa(set(A),bool,member(A,aa(product_prod(A,product_prod(code_natural,code_natural)),A,product_fst(A,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),select_weight(A,Xs),S2))),aa(list(A),set(A),set2(A),aa(list(product_prod(code_natural,A)),list(A),map(product_prod(code_natural,A),A,product_snd(code_natural,A)),Xs)))) ) ).

% select_weight_member
tff(fact_4502_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = A6 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != aa(set(B),nat,finite_card(B),A6) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_4503_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_rf(list(A),bool),Xs) ) ) ).

% transpose_transpose
tff(fact_4504_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_sc(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),filter2(list(A),aTP_Lamp_rf(list(A),bool),Xss)) ).

% transpose_aux_filter_head
tff(fact_4505_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_ua(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_4506_hd__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( aa(list(nat),nat,hd(nat),upt(I2,J2)) = I2 ) ) ).

% hd_upt
tff(fact_4507_relcomp__distrib2,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: set(product_prod(A,C)),T8: set(product_prod(A,C)),R2: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),set(product_prod(A,C))),sup_sup(set(product_prod(A,C))),S),T8),R2) = 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))),relcomp(A,C,B,S,R2)),relcomp(A,C,B,T8,R2)) ).

% relcomp_distrib2
tff(fact_4508_relcomp__distrib,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S: set(product_prod(C,B)),T8: set(product_prod(C,B))] : relcomp(A,C,B,R2,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),S),T8)) = 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))),relcomp(A,C,B,R2,S)),relcomp(A,C,B,R2,T8)) ).

% relcomp_distrib
tff(fact_4509_in__hd__or__tl__conv,axiom,
    ! [A: $tType,L: list(A),X: A] :
      ( ( L != nil(A) )
     => ( ( ( X = aa(list(A),A,hd(A),L) )
          | pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),L)))) )
      <=> pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),L))) ) ) ).

% in_hd_or_tl_conv
tff(fact_4510_rev__split__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,aa(list(A),list(A),tl(A),L))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),L)),nil(A))) = rev(A,L) ) ) ).

% rev_split_conv
tff(fact_4511_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),zip(A,B,Xs,Ys2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys2)) ) ) ) ).

% hd_zip
tff(fact_4512_relcomp__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,R6: set(product_prod(A,B)),R3: set(product_prod(A,B)),S3: set(product_prod(B,C)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R6),R3))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),bool),ord_less_eq(set(product_prod(B,C))),S3),S2))
       => pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R6,S3)),relcomp(A,B,C,R3,S2))) ) ) ).

% relcomp_mono
tff(fact_4513_relcomp_Ocases,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R3: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R3,S2)))
     => ~ ! [B4: B] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B4)),R3))
           => ~ pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),A22)),S2)) ) ) ).

% relcomp.cases
tff(fact_4514_relcomp_Osimps,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R3: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R3,S2)))
    <=> ? [A9: A,B7: B,C5: C] :
          ( ( A1 = A9 )
          & ( A22 = C5 )
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B7)),R3))
          & pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B7),C5)),S2)) ) ) ).

% relcomp.simps
tff(fact_4515_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R3: set(product_prod(A,B)),C3: C,S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
     => ( pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)),S2))
       => pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C3)),relcomp(A,B,C,R3,S2))) ) ) ).

% relcomp.relcompI
tff(fact_4516_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R3: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Xz),relcomp(A,C,B,R3,S2)))
     => ~ ! [X3: A,Y3: C,Z3: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3) )
           => ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X3),Y3)),R3))
             => ~ pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Z3)),S2)) ) ) ) ).

% relcompE
tff(fact_4517_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R3: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),C3)),relcomp(A,C,B,R3,S2)))
     => ~ ! [B4: C] :
            ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B4)),R3))
           => ~ pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C3)),S2)) ) ) ).

% relcompEpair
tff(fact_4518_length__takeWhile__le,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_takeWhile_le
tff(fact_4519_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P2,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_4520_trancl__unfold,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_trancl(A,R3) = 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))),R3),relcomp(A,A,A,transitive_trancl(A,R3),R3)) ).

% trancl_unfold
tff(fact_4521_union__comp__emptyR,axiom,
    ! [A: $tType,A6: set(product_prod(A,A)),B5: set(product_prod(A,A)),C4: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A6,B5) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,A6,C4) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,A6,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))),B5),C4)) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyR
tff(fact_4522_union__comp__emptyL,axiom,
    ! [A: $tType,A6: set(product_prod(A,A)),C4: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A6,C4) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,B5,C4) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,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))),A6),B5),C4) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyL
tff(fact_4523_not__hd__in__tl,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( X != aa(list(A),A,hd(A),Xs) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs)))) ) ) ).

% not_hd_in_tl
tff(fact_4524_trancl__Int__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,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))),inf_inf(set(product_prod(A,A))),transitive_trancl(A,R3)),S2),R3)),S2))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),S2)) ) ) ).

% trancl_Int_subset
tff(fact_4525_takeWhile__nth,axiom,
    ! [A: $tType,J2: nat,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))))
     => ( aa(nat,A,nth(A,takeWhile(A,P2,Xs)),J2) = aa(nat,A,nth(A,Xs),J2) ) ) ).

% takeWhile_nth
tff(fact_4526_nth__length__takeWhile,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
     => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))))) ) ).

% nth_length_takeWhile
tff(fact_4527_distinct__hd__tl,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( ( X = aa(list(A),A,hd(A),Xs) )
       => ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs)))) ) ) ).

% distinct_hd_tl
tff(fact_4528_sorted__hd__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ! [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,hd(A),Xs)),X5)) ) ) ) ) ).

% sorted_hd_min
tff(fact_4529_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J2: nat,P2: fun(A,bool),Xs: list(A)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_4530_less__length__takeWhile__conv,axiom,
    ! [A: $tType,I2: nat,P2: fun(A,bool),L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,L))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
        & ! [J: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
           => pp(aa(A,bool,P2,aa(nat,A,nth(A,L),J))) ) ) ) ).

% less_length_takeWhile_conv
tff(fact_4531_eq__len__takeWhile__conv,axiom,
    ! [A: $tType,I2: nat,P2: fun(A,bool),L: list(A)] :
      ( ( I2 = aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,L)) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
        & ! [J: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I2))
           => pp(aa(A,bool,P2,aa(nat,A,nth(A,L),J))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
         => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,L),I2))) ) ) ) ).

% eq_len_takeWhile_conv
tff(fact_4532_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4533_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),T5: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(B),list(A),map(B,A,F3),Xs)))
         => ( filter2(B,aa(A,fun(B,bool),aTP_Lamp_qw(fun(B,A),fun(A,fun(B,bool)),F3),T5),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_qw(fun(B,A),fun(A,fun(B,bool)),F3),T5),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_4534_ntrancl__Suc,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,N),R2) = relcomp(A,A,A,transitive_ntrancl(A,N,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))),id2(A)),R2)) ).

% ntrancl_Suc
tff(fact_4535_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L))
              & ( aa(list(B),set(B),set2(B),L) = A6 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A6) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A6) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_4536_min__ext__compat,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),R2))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R2),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R2))) ) ).

% min_ext_compat
tff(fact_4537_insort__key__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [A3: B,Xs: list(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,A3),aa(list(B),set(B),set2(B),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
           => ( ( aa(list(B),B,hd(B),filter2(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ub(B,fun(fun(B,A),fun(B,bool)),A3),F3),Xs)) = A3 )
             => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),A3),remove1(B,A3,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_4538_set__remove1__subset,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,X,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_remove1_subset
tff(fact_4539_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A3,Xs)) ) ) ).

% sorted_remove1
tff(fact_4540_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),remove1(B,X,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_4541_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),X: B,A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B))))) = remove1(B,X,sorted8670434370408473282of_set(A,B,Less_eq,F3,A6)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_4542_remove1__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( remove1(A,aa(list(A),A,hd(A),Xs),Xs) = aa(list(A),list(A),tl(A),Xs) ) ) ).

% remove1_tl
tff(fact_4543_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B),B5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),S))
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A6) = sorted8670434370408473282of_set(A,B,Less_eq,F3,B5) )
           => ( pp(aa(set(B),bool,finite_finite2(B),A6))
             => ( pp(aa(set(B),bool,finite_finite2(B),B5))
               => ( A6 = B5 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_4544_sum__list__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [X: B,Xs: list(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,X),aa(list(B),set(B),set2(B),Xs)))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,X)),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F3),remove1(B,X,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_4545_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,member(A,A3),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_pk(A,A)),A3),remove1(A,A3,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_4546_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(list(B),set(B),set2(B),sorted8670434370408473282of_set(A,B,Less_eq,F3,A6)) = A6 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_4547_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => ( aa(list(B),nat,size_size(list(B)),sorted8670434370408473282of_set(A,B,Less_eq,F3,A6)) = aa(set(B),nat,finite_card(B),A6) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_4548_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => distinct(A,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A6))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_4549_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A6))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_4550_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A6))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_4551_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A6) = nil(B) )
          <=> ( A6 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4552_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S))
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),Xs))
         => ( distinct(B,Xs)
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_4553_max__ext__compat,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),R2))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R2),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R2))) ) ).

% max_ext_compat
tff(fact_4554_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),X: B,A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)) = insort_key(A,B,Less_eq,F3,X,sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_4555_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S: set(B),F3: fun(B,A),X: B,A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)),S))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ~ pp(aa(set(B),bool,member(B,X),A6))
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A6)) = insort_key(A,B,Less_eq,F3,X,sorted8670434370408473282of_set(A,B,Less_eq,F3,A6)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_4556_extract__def,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : extract(A,P2,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_uc(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P2),Xs)),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P2),Xs)) ).

% extract_def
tff(fact_4557_length__dropWhile__le,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_dropWhile_le
tff(fact_4558_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P2,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_4559_max__ext__additive,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),R2: set(product_prod(A,A)),C4: set(A),D5: set(A)] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A6),B5)),max_ext(A,R2)))
     => ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),C4),D5)),max_ext(A,R2)))
       => pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),D5))),max_ext(A,R2))) ) ) ).

% max_ext_additive
tff(fact_4560_find__dropWhile,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : find(A,P2,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_ud(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P2),Xs)) ).

% find_dropWhile
tff(fact_4561_length__dropWhile__takeWhile,axiom,
    ! [A: $tType,X: nat,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),dropWhile(A,P2,Xs))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_dropWhile_takeWhile
tff(fact_4562_dropWhile__nth,axiom,
    ! [A: $tType,J2: nat,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),dropWhile(A,P2,Xs))))
     => ( aa(nat,A,nth(A,dropWhile(A,P2,Xs)),J2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_4563_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ( dropWhile(A,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),X),rev(A,Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),rev(A,takeWhile(A,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),X),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_4564_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ( takeWhile(A,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),X),rev(A,Xs)) = rev(A,aa(list(A),list(A),tl(A),dropWhile(A,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),X),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_4565_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R2)))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
         => ( pp(aa(set(A),bool,finite_finite2(A),A22))
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),A1))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),A22))
                        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa4)),R2)) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_4566_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R2)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
        & pp(aa(set(A),bool,finite_finite2(A),A22))
        & ( A22 != bot_bot(set(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A1))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),A22))
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R2)) ) ) ) ) ).

% max_ext.simps
tff(fact_4567_max__ext_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
       => ( ( Y6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),Y6))
                    & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa)),R2)) ) )
           => pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X6),Y6)),max_ext(A,R2))) ) ) ) ) ).

% max_ext.max_extI
tff(fact_4568_remove__rev__def,axiom,
    ! [A: $tType,X: A] : remove_rev(A,X) = aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),aa(A,fun(A,bool),fequal(A),X))) ).

% remove_rev_def
tff(fact_4569_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat)] :
      ( ( ( Xs = nil(A) )
       => ( size_list(A,F3,Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( size_list(A,F3,Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(list(A),A,hd(A),Xs))),size_list(A,F3,aa(list(A),list(A),tl(A),Xs)))) ) ) ) ).

% Nitpick.size_list_simp(1)
tff(fact_4570_max__ext__def,axiom,
    ! [A: $tType,X5: set(product_prod(A,A))] : max_ext(A,X5) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),max_extp(A,aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),X5)))) ).

% max_ext_def
tff(fact_4571_size__list__append,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A),Ys2: list(A)] : size_list(A,F3,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(A,F3,Xs)),size_list(A,F3,Ys2)) ).

% size_list_append
tff(fact_4572_size__list__estimation,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(A,nat,F3,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),size_list(A,F3,Xs))) ) ) ).

% size_list_estimation
tff(fact_4573_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),G3: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X3)),aa(A,nat,G3,X3))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),size_list(A,F3,Xs)),size_list(A,G3,Xs))) ) ).

% size_list_pointwise
tff(fact_4574_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(A,nat,F3,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),size_list(A,F3,Xs))) ) ) ).

% size_list_estimation'
tff(fact_4575_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : size_list(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),size_list(A,X,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_4576_size__list__conv__sum__list,axiom,
    ! [B: $tType,F3: fun(B,nat),Xs: list(B)] : size_list(B,F3,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,F3),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% size_list_conv_sum_list
tff(fact_4577_filter__rev__alt,axiom,
    ! [A: $tType,P2: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),P2),L) = filter2(A,P2,rev(A,L)) ).

% filter_rev_alt
tff(fact_4578_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X5: set(A),Xa: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)),X5),Xa))
    <=> pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Xa)),max_ext(A,R2))) ) ).

% max_extp_max_ext_eq
tff(fact_4579_max__extp__eq,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),X: set(A),Y: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R3),X),Y))
    <=> pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Y)),max_ext(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% max_extp_eq
tff(fact_4580_partition__rev__filter__conv,axiom,
    ! [A: $tType,P2: fun(A,bool),Yes2: list(A),No2: list(A),Xs: list(A)] : partition_rev(A,P2,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)),Yes2),No2),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),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter2(A,P2,Xs))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter2(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P2),Xs))),No2)) ).

% partition_rev_filter_conv
tff(fact_4581_map__distinct__upd__conv,axiom,
    ! [B: $tType,A: $tType,I2: nat,L: list(A),F3: fun(A,B),X: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
     => ( distinct(A,L)
       => ( list_update(B,aa(list(A),list(B),map(A,B,F3),L),I2,X) = aa(list(A),list(B),map(A,B,fun_upd(A,B,F3,aa(nat,A,nth(A,L),I2),X)),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_4582_map__of__distinct__lookup,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( aa(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),X),Y)),Ys2))),X) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_distinct_lookup
tff(fact_4583_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,X: A,X5: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_bp(A,option(B)),X,none(B)),X5) = none(B) ).

% empty_upd_none
tff(fact_4584_map__option__o__map__upd,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),M: fun(A,option(C)),A3: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),fun_upd(A,option(C),M,A3,aa(C,option(C),some(C),B2))) = fun_upd(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),M),A3,aa(B,option(B),some(B),aa(C,B,F3,B2))) ).

% map_option_o_map_upd
tff(fact_4585_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_bp(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_4586_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M,A3) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M,A3,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),ran(B,A,M)) ) ) ).

% ran_map_upd
tff(fact_4587_map__of__rev__distinct,axiom,
    ! [B: $tType,A: $tType,M: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),M))
     => ( map_of(A,B,rev(product_prod(A,B),M)) = map_of(A,B,M) ) ) ).

% map_of_rev_distinct
tff(fact_4588_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) )
      <=> pp(aa(set(product_prod(A,B)),bool,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)),Xys))) ) ) ).

% map_of_eq_Some_iff
tff(fact_4589_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X) )
      <=> pp(aa(set(product_prod(A,B)),bool,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)),Xys))) ) ) ).

% Some_eq_map_of_iff
tff(fact_4590_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( pp(aa(set(product_prod(A,B)),bool,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)),Xys)))
       => ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_4591_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),A3: B,B2: A,X: B,Y: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),M,A3,aa(A,option(A),some(A),B2)),X) = aa(A,option(A),some(A),Y) )
    <=> ( ( ( X = A3 )
          & ( B2 = Y ) )
        | ( ( X != A3 )
          & ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_4592_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,T5: fun(B,option(A)),K2: B,X: A] :
      ( ( aa(B,option(A),T5,K2) = aa(A,option(A),some(A),X) )
     => ( fun_upd(B,option(A),T5,K2,aa(A,option(A),some(A),X)) = T5 ) ) ).

% map_upd_triv
tff(fact_4593_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,X: B,N: fun(A,option(B)),Y: B] :
      ( ( fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),X)) = fun_upd(A,option(B),N,A3,aa(B,option(B),some(B),Y)) )
     => ( X = Y ) ) ).

% map_upd_eqD1
tff(fact_4594_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C3: B,P2: fun(fun(A,B),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_ue(fun(A,B),fun(B,fun(A,bool)),F3),C3))))
     => ( pp(aa(fun(A,B),bool,P2,aTP_Lamp_uf(B,fun(A,B),C3)))
       => ( ! [A5: A,B4: B,F: fun(A,B)] :
              ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ug(B,fun(fun(A,B),fun(A,bool)),C3),F))))
             => ( ( aa(A,B,F,A5) = C3 )
               => ( ( B4 != C3 )
                 => ( pp(aa(fun(A,B),bool,P2,F))
                   => pp(aa(fun(A,B),bool,P2,fun_upd(A,B,F,A5,B4))) ) ) ) )
         => pp(aa(fun(A,B),bool,P2,F3)) ) ) ) ).

% finite_update_induct
tff(fact_4595_map__of__zip__upd,axiom,
    ! [A: $tType,B: $tType,Ys2: list(B),Xs: list(A),Zs: list(B),X: A,Y: B,Z4: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys2) = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
       => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),X,aa(B,option(B),some(B),Y)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X,aa(B,option(B),some(B),Z4)) )
           => ( map_of(A,B,zip(A,B,Xs,Ys2)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_4596_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,T5: fun(A,option(B)),K2: A,X: B] :
      ~ ! [X3: A] : aa(A,option(B),fun_upd(A,option(B),T5,K2,aa(B,option(B),some(B),X)),X3) = none(B) ).

% map_upd_nonempty
tff(fact_4597_map__of_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),Ps: list(product_prod(A,B))] : map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),P3),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P3),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P3))) ).

% map_of.simps(2)
tff(fact_4598_map__of__None__filterD,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),X: B,P2: fun(product_prod(B,A),bool)] :
      ( ( aa(B,option(A),map_of(B,A,Xs),X) = none(A) )
     => ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),P2,Xs)),X) = none(A) ) ) ).

% map_of_None_filterD
tff(fact_4599_partition__rev_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),Yes2: list(A),No2: list(A),X: A,Xs: list(A)] : partition_rev(A,P2,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)),Yes2),No2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = partition_rev(A,P2,if(product_prod(list(A),list(A)),aa(A,bool,P2,X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Yes2)),No2),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)),Yes2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),No2))),Xs) ).

% partition_rev.simps(2)
tff(fact_4600_partition__rev_Osimps_I1_J,axiom,
    ! [A: $tType,P2: fun(A,bool),Yes2: list(A),No2: list(A)] : partition_rev(A,P2,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)),Yes2),No2),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)),Yes2),No2) ).

% partition_rev.simps(1)
tff(fact_4601_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K2: A,X: B,L: list(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),X)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L)))
     => ? [X3: B] : aa(A,option(B),map_of(A,B,L),K2) = aa(B,option(B),some(B),X3) ) ).

% weak_map_of_SomeI
tff(fact_4602_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K2: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Y) )
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),Y)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ) ).

% map_of_SomeD
tff(fact_4603_map__of__Cons__code_I2_J,axiom,
    ! [C: $tType,B: $tType,L: B,K2: B,V2: C,Ps: list(product_prod(B,C))] :
      ( ( ( L = K2 )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V2)),Ps)),K2) = aa(C,option(C),some(C),V2) ) )
      & ( ( L != K2 )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V2)),Ps)),K2) = aa(B,option(C),map_of(B,C,Ps),K2) ) ) ) ).

% map_of_Cons_code(2)
tff(fact_4604_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K2: B,Z4: A,P2: fun(B,fun(A,bool))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Z4) )
     => ( pp(aa(A,bool,aa(B,fun(A,bool),P2,K2),Z4))
       => ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P2),Xs)),K2) = aa(A,option(A),some(A),Z4) ) ) ) ).

% map_of_filter_in
tff(fact_4605_map__of__distinct__upd3,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B,Y7: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),X),Y)),Ys2))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),X),Y7)),Ys2))),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd3
tff(fact_4606_map__of__distinct__upd2,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),X),Y)),Ys2))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys2)),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd2
tff(fact_4607_map__of__distinct__upd4,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Ys2: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys2))))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys2)) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),X),Y)),Ys2))),X,none(B)) ) ) ) ).

% map_of_distinct_upd4
tff(fact_4608_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
      <=> ? [Y4: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),X) = aa(B,option(B),some(B),Y4) ) ) ).

% map_of_zip_is_Some
tff(fact_4609_partition__rev_Oelims,axiom,
    ! [A: $tType,X: fun(A,bool),Xa2: product_prod(list(A),list(A)),Xb2: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,X,Xa2,Xb2) = Y )
     => ( ! [Yes: list(A),No: list(A)] :
            ( ( Xa2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
           => ( ( Xb2 = nil(A) )
             => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) ) ) )
       => ~ ! [Yes: list(A),No: list(A)] :
              ( ( Xa2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
             => ! [X3: A,Xs2: list(A)] :
                  ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( Y != partition_rev(A,X,if(product_prod(list(A),list(A)),aa(A,bool,X,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),No))),Xs2) ) ) ) ) ) ).

% partition_rev.elims
tff(fact_4610_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,B),X5: A] :
      ( ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X5) = aa(B,option(B),some(B),aa(A,B,F3,X5)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X5) = none(B) ) ) ) ).

% map_of_zip_map
tff(fact_4611_map__of__Some__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K2: B,V2: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),V2) )
     => ? [Ys5: list(product_prod(B,A)),Zs2: list(product_prod(B,A))] :
          ( ( Xs = aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(list(product_prod(B,A)),fun(list(product_prod(B,A)),list(product_prod(B,A))),append(product_prod(B,A)),Ys5),aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),V2)),Zs2)) )
          & ( aa(B,option(A),map_of(B,A,Ys5),K2) = none(A) ) ) ) ).

% map_of_Some_split
tff(fact_4612_map__to__set__map__of,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
     => ( map_to_set(A,B,map_of(A,B,L)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) ) ) ).

% map_to_set_map_of
tff(fact_4613_map__of__map__to__set,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B)),M: fun(A,option(B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
     => ( ( map_of(A,B,L) = M )
      <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) = map_to_set(A,B,M) ) ) ) ).

% map_of_map_to_set
tff(fact_4614_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),I2: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( distinct(A,Xs)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys2)))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),aa(nat,A,nth(A,Xs),I2)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),I2)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_4615_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),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_rw(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_4616_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_uh(list(product_prod(A,B)),fun(A,fun(B,bool)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_4617_map__of__Some__filter__not__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K2: B,V2: A,P2: fun(product_prod(B,A),bool)] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),V2) )
     => ( ~ pp(aa(product_prod(B,A),bool,P2,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),V2)))
       => ( distinct(B,aa(list(product_prod(B,A)),list(B),map(product_prod(B,A),B,product_fst(B,A)),Xs))
         => ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),P2,Xs)),K2) = none(A) ) ) ) ) ).

% map_of_Some_filter_not_in
tff(fact_4618_partition__rev_Opelims,axiom,
    ! [A: $tType,X: fun(A,bool),Xa2: product_prod(list(A),list(A)),Xb2: list(A),Y: product_prod(list(A),list(A))] :
      ( ( partition_rev(A,X,Xa2,Xb2) = Y )
     => ( pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xa2),Xb2))))
       => ( ! [Yes: list(A),No: list(A)] :
              ( ( Xa2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
             => ( ( Xb2 = nil(A) )
               => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
                 => ~ pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A))))) ) ) )
         => ~ ! [Yes: list(A),No: list(A)] :
                ( ( Xa2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
               => ! [X3: A,Xs2: list(A)] :
                    ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ( ( Y = partition_rev(A,X,if(product_prod(list(A),list(A)),aa(A,bool,X,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),No))),Xs2) )
                     => ~ pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))))) ) ) ) ) ) ) ).

% partition_rev.pelims
tff(fact_4619_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),M: fun(A,option(B)),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2)))
     => ( map_upds(A,B,M,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Ys2) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys2),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_4620_quicksort__by__rel_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa2),Xb2) = Y )
     => ( ( ( Xb2 = nil(A) )
         => ( Y != Xa2 ) )
       => ~ ! [X3: A,Xs2: list(A)] :
              ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ( Y != aa(product_prod(list(A),list(A)),list(A),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(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_ui(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa2),X3)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)) ) ) ) ) ).

% quicksort_by_rel.elims
tff(fact_4621_set__quicksort__by__rel,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).

% set_quicksort_by_rel
tff(fact_4622_quicksort__by__rel__permutes,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : mset(A,aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs)) = mset(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).

% quicksort_by_rel_permutes
tff(fact_4623_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,M: fun(A,option(B)),Ys2: list(B),Y: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( map_upds(A,B,M,Xs,list_update(B,Ys2,I2,Y)) = map_upds(A,B,M,Xs,Ys2) ) ) ).

% map_upds_list_update2_drop
tff(fact_4624_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,As: list(A),B2: B,Bs: list(B)] : map_upds(A,B,M,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),B2)),As,Bs) ).

% map_upds_Cons
tff(fact_4625_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A3: A,As: list(A),M: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),aa(list(A),set(A),set2(A),As)))
     => ( map_upds(A,B,fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),B2)),As,Bs) = fun_upd(A,option(B),map_upds(A,B,M,As,Bs),A3,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_4626_quicksort__by__rel_Osimps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),nil(A)) = Sl2 ).

% quicksort_by_rel.simps(1)
tff(fact_4627_quicksort__by__rel__remove__acc,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R2,nil(A)),Xs)),Sl2) ).

% quicksort_by_rel_remove_acc
tff(fact_4628_quicksort__by__rel__remove__acc__guared,axiom,
    ! [A: $tType,Sl2: list(A),R2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( ( Sl2 != nil(A) )
     => ( aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R2,nil(A)),Xs)),Sl2) ) ) ).

% quicksort_by_rel_remove_acc_guared
tff(fact_4629_sorted__wrt__quicksort__by__rel,axiom,
    ! [X9: $tType,R2: fun(X9,fun(X9,bool)),Xs: list(X9)] :
      ( ! [X3: X9,Y3: X9] :
          ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,X3),Y3))
          | pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,Y3),X3)) )
     => ( ! [X3: X9,Y3: X9,Z3: X9] :
            ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,X3),Y3))
           => ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,Y3),Z3))
             => pp(aa(X9,bool,aa(X9,fun(X9,bool),R2,X3),Z3)) ) )
       => sorted_wrt(X9,R2,aa(list(X9),list(X9),quicksort_by_rel(X9,R2,nil(X9)),Xs)) ) ) ).

% sorted_wrt_quicksort_by_rel
tff(fact_4630_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) = aa(list(product_prod(A,B)),fun(A,option(B)),aa(fun(A,option(B)),fun(list(product_prod(A,B)),fun(A,option(B))),foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_uk(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_4631_sorted__quicksort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),quicksort_by_rel(A,ord_less_eq(A),nil(A)),Xs)) ) ).

% sorted_quicksort_by_rel
tff(fact_4632_sort__quicksort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_pk(A,A)) = quicksort_by_rel(A,ord_less_eq(A),nil(A)) ) ) ).

% sort_quicksort_by_rel
tff(fact_4633_quicksort__by__rel_Osimps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),X: A,Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),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(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_ui(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R2),Sl2),X)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R2),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ).

% quicksort_by_rel.simps(2)
tff(fact_4634_quicksort__by__rel_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: list(A),Xb2: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),quicksort_by_rel(A,X,Xa2),Xb2) = Y )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),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)),Xa2),Xb2))))
       => ( ( ( Xb2 = nil(A) )
           => ( ( Y = Xa2 )
             => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),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)),Xa2),nil(A))))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( Xb2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( ( Y = aa(product_prod(list(A),list(A)),list(A),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(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_ui(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X),Xa2),X3)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)) )
                 => ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X),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)),Xa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))))) ) ) ) ) ) ).

% quicksort_by_rel.pelims
tff(fact_4635_quicksort__by__rel_Opsimps_I2_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A),X: A,Xs: list(A)] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R2),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)),Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))))
     => ( aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),list(A),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(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_ui(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R2),Sl2),X)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R2),X),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ) ) ).

% quicksort_by_rel.psimps(2)
tff(fact_4636_quicksort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),A22: list(A),P2: fun(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)))] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),A0),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)),A1),A22))))
     => ( ! [R7: fun(A,fun(A,bool)),Sl: list(A)] :
            ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P2,R7),Sl),nil(A))) )
       => ( ! [R7: fun(A,fun(A,bool)),Sl: list(A),X3: A,Xs2: list(A)] :
              ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)))))
             => ( ! [Xa: product_prod(list(A),list(A)),Xb: list(A),Y5: list(A)] :
                    ( ( Xa = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R7),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2) )
                   => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y5) = Xa )
                     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P2,R7),Sl),Y5)) ) )
               => ( ! [Xa: product_prod(list(A),list(A)),Xb: list(A),Y5: list(A)] :
                      ( ( Xa = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R7),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2) )
                     => ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y5) = Xa )
                       => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P2,R7),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),quicksort_by_rel(A,R7,Sl),Y5))),Xb)) ) )
                 => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P2,R7),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ) )
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P2,A0),A1),A22)) ) ) ) ).

% quicksort_by_rel.pinduct
tff(fact_4637_quicksort__by__rel_Opsimps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),Sl2: list(A)] :
      ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R2),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)),Sl2),nil(A)))))
     => ( aa(list(A),list(A),quicksort_by_rel(A,R2,Sl2),nil(A)) = Sl2 ) ) ).

% quicksort_by_rel.psimps(1)
tff(fact_4638_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K2: A,V2: B] : graph(A,B,fun_upd(A,option(B),M,K2,aa(B,option(B),some(B),V2))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(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),K2),V2)),graph(A,B,fun_upd(A,option(B),M,K2,none(B)))) ).

% graph_map_upd
tff(fact_4639_map__of__distinct__upd,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(product_prod(A,B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
     => ( map_add(A,B,fun_upd(A,option(B),aTP_Lamp_bp(A,option(B)),X,aa(B,option(B),some(B),Y)),map_of(A,B,Xs)) = fun_upd(A,option(B),map_of(A,B,Xs),X,aa(B,option(B),some(B),Y)) ) ) ).

% map_of_distinct_upd
tff(fact_4640_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),X6: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S))
     => ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert(product_prod(C,A)),X),bot_bot(set(product_prod(C,A)))),S)),X6) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ul(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X6,S) ) ) ).

% insert_relcomp_union_fold
tff(fact_4641_map__add__find__right,axiom,
    ! [B: $tType,A: $tType,N: fun(B,option(A)),K2: B,Xx: A,M: fun(B,option(A))] :
      ( ( aa(B,option(A),N,K2) = aa(A,option(A),some(A),Xx) )
     => ( aa(B,option(A),map_add(B,A,M,N),K2) = aa(A,option(A),some(A),Xx) ) ) ).

% map_add_find_right
tff(fact_4642_empty__eq__map__add__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( ( aTP_Lamp_bp(A,option(B)) = map_add(A,B,F3,G3) )
    <=> ( ! [X4: A] : aa(A,option(B),F3,X4) = none(B)
        & ! [X4: A] : aa(A,option(B),G3,X4) = none(B) ) ) ).

% empty_eq_map_add_iff
tff(fact_4643_map__add__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,M,aTP_Lamp_bp(A,option(B))) = M ).

% map_add_empty
tff(fact_4644_empty__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,aTP_Lamp_bp(A,option(B)),M) = M ).

% empty_map_add
tff(fact_4645_map__add__upd,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),G3: fun(A,option(B)),X: A,Y: B] : map_add(A,B,F3,fun_upd(A,option(B),G3,X,aa(B,option(B),some(B),Y))) = fun_upd(A,option(B),map_add(A,B,F3,G3),X,aa(B,option(B),some(B),Y)) ).

% map_add_upd
tff(fact_4646_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_bp(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_4647_map__add__left__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),K2: B,G3: fun(B,option(A))] :
      ( ( aa(B,option(A),F3,K2) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F3,G3),K2) = aa(B,option(A),G3,K2) ) ) ).

% map_add_left_None
tff(fact_4648_map__add__find__left,axiom,
    ! [A: $tType,B: $tType,G3: fun(B,option(A)),K2: B,F3: fun(B,option(A))] :
      ( ( aa(B,option(A),G3,K2) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F3,G3),K2) = aa(B,option(A),F3,K2) ) ) ).

% map_add_find_left
tff(fact_4649_map__add__first__le,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M: fun(A,option(B)),M8: fun(A,option(B)),N: fun(A,option(B))] :
          ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M8))
         => pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N)),map_add(A,B,M8,N))) ) ) ).

% map_add_first_le
tff(fact_4650_map__add__SomeD,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K2: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K2) = aa(A,option(A),some(A),X) )
     => ( ( aa(B,option(A),N,K2) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K2) = none(A) )
          & ( aa(B,option(A),M,K2) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_SomeD
tff(fact_4651_map__add__Some__iff,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K2: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K2) = aa(A,option(A),some(A),X) )
    <=> ( ( aa(B,option(A),N,K2) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K2) = none(A) )
          & ( aa(B,option(A),M,K2) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_Some_iff
tff(fact_4652_in__graphD,axiom,
    ! [A: $tType,B: $tType,K2: A,V2: B,M: fun(A,option(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V2)),graph(A,B,M)))
     => ( aa(A,option(B),M,K2) = aa(B,option(B),some(B),V2) ) ) ).

% in_graphD
tff(fact_4653_in__graphI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K2: B,V2: A] :
      ( ( aa(B,option(A),M,K2) = aa(A,option(A),some(A),V2) )
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),V2)),graph(B,A,M))) ) ).

% in_graphI
tff(fact_4654_map__add__def,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X5: A] : aa(A,option(B),map_add(A,B,M1,M22),X5) = case_option(option(B),B,aa(A,option(B),M1,X5),some(B),aa(A,option(B),M22,X5)) ).

% map_add_def
tff(fact_4655_union__fold__insert,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = finite_fold(A,set(A),insert(A),B5,A6) ) ) ).

% union_fold_insert
tff(fact_4656_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A6: set(B)] : groups7311177749621191930dd_sum(B,A,G3,A6) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G3),zero_zero(A),A6) ) ).

% sum.eq_fold
tff(fact_4657_map__add__map__of__foldr,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,M,map_of(A,B,Ps)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(product_prod(A,B),fun(A,option(B)),aa(fun(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),product_case_prod(A,B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_um(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps),M) ).

% map_add_map_of_foldr
tff(fact_4658_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R2))
     => ( pp(aa(set(product_prod(B,C)),bool,finite_finite2(product_prod(B,C)),S))
       => ( relcomp(A,B,C,R2,S) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_uo(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S)),bot_bot(set(product_prod(A,C))),R2) ) ) ) ).

% relcomp_fold
tff(fact_4659_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K2: A] : graph(A,B,fun_upd(A,option(B),M,K2,none(B))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),bool),aTP_Lamp_up(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M),K2)) ).

% graph_fun_upd_None
tff(fact_4660_map__of__concat,axiom,
    ! [B: $tType,A: $tType,Xss: list(list(product_prod(A,B)))] : map_of(A,B,concat(product_prod(A,B),Xss)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(list(product_prod(A,B)),fun(A,option(B)),aTP_Lamp_uq(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Xss),aTP_Lamp_bp(A,option(B))) ).

% map_of_concat
tff(fact_4661_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),X: product_prod(C,A),R2: set(product_prod(C,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S))
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert(product_prod(C,A)),X),R2),S) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ul(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R2,S),S) ) ) ).

% insert_relcomp_fold
tff(fact_4662_Set__filter__fold,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( filter3(A,P2,A6) = finite_fold(A,set(A),aTP_Lamp_ur(fun(A,bool),fun(A,fun(set(A),set(A))),P2),bot_bot(set(A)),A6) ) ) ).

% Set_filter_fold
tff(fact_4663_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S))
     => finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ut(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_4664_Id__on__fold,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( id_on(A,A6) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_uu(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A6) ) ) ).

% Id_on_fold
tff(fact_4665_member__filter,axiom,
    ! [A: $tType,X: A,P2: fun(A,bool),A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),filter3(A,P2,A6)))
    <=> ( pp(aa(set(A),bool,member(A,X),A6))
        & pp(aa(A,bool,P2,X)) ) ) ).

% member_filter
tff(fact_4666_Id__onI,axiom,
    ! [A: $tType,A3: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id_on(A,A6))) ) ).

% Id_onI
tff(fact_4667_Set_Ofilter__def,axiom,
    ! [A: $tType,P2: fun(A,bool),A6: set(A)] : filter3(A,P2,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_uv(fun(A,bool),fun(set(A),fun(A,bool)),P2),A6)) ).

% Set.filter_def
tff(fact_4668_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_uw(fun(B,B),fun(A,fun(B,B)),F3)) ).

% comp_fun_commute_const
tff(fact_4669_Id__on__iff,axiom,
    ! [A: $tType,X: A,Y: A,A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),id_on(A,A6)))
    <=> ( ( X = Y )
        & pp(aa(set(A),bool,member(A,X),A6)) ) ) ).

% Id_on_iff
tff(fact_4670_Id__on__eqI,axiom,
    ! [A: $tType,A3: A,B2: A,A6: set(A)] :
      ( ( A3 = B2 )
     => ( pp(aa(set(A),bool,member(A,A3),A6))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id_on(A,A6))) ) ) ).

% Id_on_eqI
tff(fact_4671_Id__onE,axiom,
    ! [A: $tType,C3: product_prod(A,A),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),C3),id_on(A,A6)))
     => ~ ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => ( C3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ) ).

% Id_onE
tff(fact_4672_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P2: fun(A,bool)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_ur(fun(A,bool),fun(A,fun(set(A),set(A))),P2)) ).

% comp_fun_commute_filter_fold
tff(fact_4673_Id__on__def_H,axiom,
    ! [A: $tType,A6: fun(A,bool)] : id_on(A,aa(fun(A,bool),set(A),collect(A),A6)) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ux(fun(A,bool),fun(A,fun(A,bool)),A6))) ).

% Id_on_def'
tff(fact_4674_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_uy(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_4675_comp__fun__commute_Ofoldl__f__commute,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),A3: A,B2: B,Xs: list(A)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(B,B,aa(A,fun(B,B),F3,A3),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_uz(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),B2),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_uz(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),aa(B,B,aa(A,fun(B,B),F3,A3),B2)),Xs) ) ) ).

% comp_fun_commute.foldl_f_commute
tff(fact_4676_card_Oeq__fold,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),nat,finite_card(A),A6) = finite_fold(A,nat,aTP_Lamp_rb(A,fun(nat,nat)),zero_zero(nat),A6) ).

% card.eq_fold
tff(fact_4677_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A6) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),nil(A),A6) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_4678_comp__fun__commute_Ofoldr__conv__foldl,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),Xs: list(A),A3: B] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(B,B,foldr(A,B,F3,Xs),A3) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_uz(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),A3),Xs) ) ) ).

% comp_fun_commute.foldr_conv_foldl
tff(fact_4679_comp__fun__commute__Image__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A)] : finite6289374366891150609ommute(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_va(set(A),fun(A,fun(B,fun(set(B),set(B)))),S))) ).

% comp_fun_commute_Image_fold
tff(fact_4680_inter__Set__filter,axiom,
    ! [A: $tType,B5: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = filter3(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6),B5) ) ) ).

% inter_Set_filter
tff(fact_4681_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_vc(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B5)) ) ).

% comp_fun_commute_product_fold
tff(fact_4682_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)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_ox(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_4683_Misc_Oran__distinct,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( ran(A,B,map_of(A,B,Al)) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).

% Misc.ran_distinct
tff(fact_4684_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),A6: set(A),B5: set(A),Z4: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),S))
         => ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = finite_fold(A,B,F3,finite_fold(A,B,F3,Z4,A6),B5) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_4685_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_4686_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),X: B,A6: set(B)] :
      ( ( B2 = aa(B,A,F3,X) )
     => ( pp(aa(set(B),bool,member(B,X),A6))
       => pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),A6))) ) ) ).

% image_eqI
tff(fact_4687_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_cc(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_4688_image__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_4689_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F3),A6) )
    <=> ( A6 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_4690_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A6) = bot_bot(set(A)) )
    <=> ( A6 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_4691_image__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: B,B5: set(B)] : aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F3,A3)),aa(set(B),set(A),image2(B,A,F3),B5)) ).

% image_insert
tff(fact_4692_insert__image,axiom,
    ! [B: $tType,A: $tType,X: A,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,member(A,X),A6))
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,X)),aa(set(A),set(B),image2(A,B,F3),A6)) = aa(set(A),set(B),image2(A,B,F3),A6) ) ) ).

% insert_image
tff(fact_4693_image__update,axiom,
    ! [B: $tType,A: $tType,X: A,A6: set(A),F3: fun(A,B),N: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),A6))
     => ( aa(set(A),set(B),image2(A,B,fun_upd(A,B,F3,X,N)),A6) = aa(set(A),set(B),image2(A,B,F3),A6) ) ) ).

% image_update
tff(fact_4694_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S) = S ) ).

% image_add_0
tff(fact_4695_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K2)),set_or1337092689740270186AtMost(A,I2,J2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastAtMost
tff(fact_4696_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K2)),set_or7035219750837199246ssThan(A,I2,J2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastLessThan
tff(fact_4697_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C3: A,A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C3)),aa(A,set(A),set_ord_atMost(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)) ) ).

% image_add_atMost
tff(fact_4698_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C3: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C3)),set_or3652927894154168847AtMost(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_4699_img__fst,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
     => pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S))) ) ).

% img_fst
tff(fact_4700_img__snd,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),S))
     => pp(aa(set(B),bool,member(B,B2),aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S))) ) ).

% img_snd
tff(fact_4701_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_vd(A,fun(A,A),K2)),set_or1337092689740270186AtMost(A,I2,J2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastAtMost'
tff(fact_4702_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ve(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_4703_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_vd(A,fun(A,A),K2)),set_or7035219750837199246ssThan(A,I2,J2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastLessThan'
tff(fact_4704_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P2: fun(B,bool),F3: fun(B,A),G3: fun(B,A),S: set(B)] : aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_vf(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P2),F3),G3)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,bool),set(B),collect(B),P2)))),aa(set(B),set(A),image2(B,A,G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_vg(fun(B,bool),fun(B,bool),P2))))) ).

% if_image_distrib
tff(fact_4705_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A6: set(product_prod(A,B)),F3: fun(A,fun(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),A6))
     => pp(aa(set(C),bool,member(C,aa(B,C,aa(A,fun(B,C),F3,A3),B2)),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)),A6))) ) ).

% pair_imageI
tff(fact_4706_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
         => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4707_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_vh(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4708_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),F3: fun(A,B),B5: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
         => pp(aa(set(B),bool,member(B,aa(A,B,F3,X3)),B5)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(fun(A,bool),set(A),collect(A),P2))),B5)) ) ).

% image_Collect_subsetI
tff(fact_4709_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),P2: fun(set(A),bool)] :
      ( ! [B11: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F3),A6)))
         => pp(aa(set(A),bool,P2,B11)) )
    <=> ! [B11: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B11),A6))
         => pp(aa(set(A),bool,P2,aa(set(B),set(A),image2(B,A,F3),B11))) ) ) ).

% all_subset_image
tff(fact_4710_image__mono,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),aa(set(A),set(B),image2(A,B,F3),B5))) ) ).

% image_mono
tff(fact_4711_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,B),B5: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(set(B),bool,member(B,aa(A,B,F3,X3)),B5)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),B5)) ) ).

% image_subsetI
tff(fact_4712_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F3),A6)))
     => ~ ! [C8: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C8),A6))
           => ( B5 != aa(set(B),set(A),image2(B,A,F3),C8) ) ) ) ).

% subset_imageE
tff(fact_4713_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A6)),B5))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
         => pp(aa(set(A),bool,member(A,aa(B,A,F3,X4)),B5)) ) ) ).

% image_subset_iff
tff(fact_4714_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F3),A6)))
    <=> ? [AA: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),AA),A6))
          & ( B5 = aa(set(B),set(A),image2(B,A,F3),AA) ) ) ) ).

% subset_image_iff
tff(fact_4715_comp__fun__commute__plus__mset,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(multiset(A),multiset(A),plus_plus(multiset(A))) ).

% comp_fun_commute_plus_mset
tff(fact_4716_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(A,C),G3: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_vi(C,set(B))),F3) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image2(D,B,G3)),aTP_Lamp_vj(A,set(D))) ).

% empty_natural
tff(fact_4717_Inf_OINF__identity__eq,axiom,
    ! [A: $tType,Inf: fun(set(A),A),A6: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),image2(A,A,aTP_Lamp_cc(A,A)),A6)) = aa(set(A),A,Inf,A6) ).

% Inf.INF_identity_eq
tff(fact_4718_Sup_OSUP__identity__eq,axiom,
    ! [A: $tType,Sup: fun(set(A),A),A6: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),image2(A,A,aTP_Lamp_cc(A,A)),A6)) = aa(set(A),A,Sup,A6) ).

% Sup.SUP_identity_eq
tff(fact_4719_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),A6)))
     => ~ ! [X3: B] :
            ( ( B2 = aa(B,A,F3,X3) )
           => ~ pp(aa(set(B),bool,member(B,X3),A6)) ) ) ).

% imageE
tff(fact_4720_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),A6: set(C)] : aa(set(B),set(A),image2(B,A,F3),aa(set(C),set(B),image2(C,B,G3),A6)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_vk(fun(B,A),fun(fun(C,B),fun(C,A)),F3),G3)),A6) ).

% image_image
tff(fact_4721_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B),P2: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_vl(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F3),A6),P2)) = aa(set(B),set(A),image2(B,A,F3),aa(fun(B,bool),set(B),collect(B),aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_vm(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F3),A6),P2))) ).

% Compr_image_eq
tff(fact_4722_imageI,axiom,
    ! [B: $tType,A: $tType,X: A,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,member(A,X),A6))
     => pp(aa(set(B),bool,member(B,aa(A,B,F3,X)),aa(set(A),set(B),image2(A,B,F3),A6))) ) ).

% imageI
tff(fact_4723_image__iff,axiom,
    ! [A: $tType,B: $tType,Z4: A,F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,Z4),aa(set(B),set(A),image2(B,A,F3),A6)))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
          & ( Z4 = aa(B,A,F3,X4) ) ) ) ).

% image_iff
tff(fact_4724_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),P2: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),aa(set(B),set(A),image2(B,A,F3),A6)))
          & pp(aa(A,bool,P2,X5)) )
     => ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A6))
          & pp(aa(A,bool,P2,aa(B,A,F3,X3))) ) ) ).

% bex_imageD
tff(fact_4725_image__cong,axiom,
    ! [B: $tType,A: $tType,M6: set(A),N7: set(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( M6 = N7 )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),N7))
           => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) )
       => ( aa(set(A),set(B),image2(A,B,F3),M6) = aa(set(A),set(B),image2(A,B,G3),N7) ) ) ) ).

% image_cong
tff(fact_4726_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),P2: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(set(B),set(A),image2(B,A,F3),A6)))
         => pp(aa(A,bool,P2,X3)) )
     => ! [X5: B] :
          ( pp(aa(set(B),bool,member(B,X5),A6))
         => pp(aa(A,bool,P2,aa(B,A,F3,X5))) ) ) ).

% ball_imageD
tff(fact_4727_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,X: A,A6: set(A),B2: B,F3: fun(A,B)] :
      ( pp(aa(set(A),bool,member(A,X),A6))
     => ( ( B2 = aa(A,B,F3,X) )
       => pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image2(A,B,F3),A6))) ) ) ).

% rev_image_eqI
tff(fact_4728_image__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),B5: set(B)] : aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F3),A6)),aa(set(B),set(A),image2(B,A,F3),B5)) ).

% image_Un
tff(fact_4729_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(set(A),set(B),image2(A,B,F3),A6)))
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
            & ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_vn(set(A),fun(fun(A,B),fun(A,fun(A,bool))),A6),F3),X3)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_4730_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => finite4664212375090638736ute_on(A,B,S,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_uy(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_4731_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),P2: fun(set(A),bool)] :
      ( ! [B11: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),B11))
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F3),A6))) )
         => pp(aa(set(A),bool,P2,B11)) )
    <=> ! [B11: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),B11))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B11),A6)) )
         => pp(aa(set(A),bool,P2,aa(set(B),set(A),image2(B,A,F3),B11))) ) ) ).

% all_finite_subset_image
tff(fact_4732_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),P2: fun(set(A),bool)] :
      ( ? [B11: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B11))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F3),A6)))
          & pp(aa(set(A),bool,P2,B11)) )
    <=> ? [B11: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B11))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B11),A6))
          & pp(aa(set(A),bool,P2,aa(set(B),set(A),image2(B,A,F3),B11))) ) ) ).

% ex_finite_subset_image
tff(fact_4733_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F3),A6)))
       => ? [C8: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C8),A6))
            & pp(aa(set(B),bool,finite_finite2(B),C8))
            & ( B5 = aa(set(B),set(A),image2(B,A,F3),C8) ) ) ) ) ).

% finite_subset_image
tff(fact_4734_finite__surj,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F3),A6)))
       => pp(aa(set(B),bool,finite_finite2(B),B5)) ) ) ).

% finite_surj
tff(fact_4735_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T5: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S2)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T5)) ) ).

% translation_Int
tff(fact_4736_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T5: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S2)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T5)) ) ).

% translation_diff
tff(fact_4737_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,X: A,S: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)))
     => ~ ! [Y3: B] : ~ pp(aa(set(product_prod(A,B)),bool,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)),S)) ) ).

% in_fst_imageE
tff(fact_4738_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,S: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,member(A,Y),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S)))
     => ~ ! [X3: B] : ~ pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y)),S)) ) ).

% in_snd_imageE
tff(fact_4739_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),B5: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image2(B,A,F3),A6)),aa(set(B),set(A),image2(B,A,F3),B5)))) ).

% image_Int_subset
tff(fact_4740_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),B5: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),image2(B,A,F3),A6)),aa(set(B),set(A),image2(B,A,F3),B5))),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5)))) ).

% image_diff_subset
tff(fact_4741_set__oo__map__alt,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X5: list(A)] : aa(list(A),set(B),aa(fun(list(A),list(B)),fun(list(A),set(B)),comp(list(B),set(B),list(A),set2(B)),map(A,B,F3)),X5) = aa(set(A),set(B),image2(A,B,F3),aa(list(A),set(A),set2(A),X5)) ).

% set_oo_map_alt
tff(fact_4742_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T5: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T5)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T5)) ) ).

% translation_Compl
tff(fact_4743_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_pk(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_4744_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A6: set(A),C3: B] :
      ( pp(aa(set(A),bool,member(A,X),A6))
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_uf(B,fun(A,B),C3)),A6) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),C3),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_4745_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,A6: set(B),C3: A] :
      ( ( ( A6 = bot_bot(set(B)) )
       => ( aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_or(A,fun(B,A)),C3)),A6) = bot_bot(set(A)) ) )
      & ( ( A6 != bot_bot(set(B)) )
       => ( aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_or(A,fun(B,A)),C3)),A6) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),bot_bot(set(A))) ) ) ) ).

% image_constant_conv
tff(fact_4746_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T5: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),T5)) ) ).

% translation_subtract_Int
tff(fact_4747_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S2: set(A),T5: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),S2)),aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),T5)) ) ).

% translation_subtract_diff
tff(fact_4748_sum_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),H: fun(B,A),G3: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( groups7311177749621191930dd_sum(B,A,H,S) = groups7311177749621191930dd_sum(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_vq(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S),H),G3),aa(set(B),set(C),image2(B,C,G3),S)) ) ) ) ).

% sum.image_gen
tff(fact_4749_prod_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),H: fun(B,A),G3: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( groups7121269368397514597t_prod(B,A,H,S) = groups7121269368397514597t_prod(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_vr(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S),H),G3),aa(set(B),set(C),image2(B,C,G3),S)) ) ) ) ).

% prod.image_gen
tff(fact_4750_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T5: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T5)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),T5)) ) ).

% translation_subtract_Compl
tff(fact_4751_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B),X: A] :
      ( ( A6 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( pp(aa(set(A),bool,member(A,Y3),A6))
           => ( aa(A,B,F3,Y3) = aa(A,B,F3,X) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F3),A6)) = aa(A,B,F3,X) ) ) ) ).

% the_elem_image_unique
tff(fact_4752_ran__map__option,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),M: fun(B,option(C))] : ran(B,A,aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_vs(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),F3),M)) = aa(set(C),set(A),image2(C,A,F3),ran(B,C,M)) ).

% ran_map_option
tff(fact_4753_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(A,B)),B5: set(A),X: A,Y: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A6)),B5))
     => ( pp(aa(set(product_prod(A,B)),bool,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)),A6))
       => pp(aa(set(A),bool,member(A,X),B5)) ) ) ).

% fst_image_mp
tff(fact_4754_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(B,A)),B5: set(A),X: B,Y: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A6)),B5))
     => ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)),A6))
       => pp(aa(set(A),bool,member(A,Y),B5)) ) ) ).

% snd_image_mp
tff(fact_4755_card__image__le,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F3),A6))),aa(set(A),nat,finite_card(A),A6))) ) ).

% card_image_le
tff(fact_4756_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(B),T8: set(C),G3: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T8))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G3),S)),T8))
             => ( groups7311177749621191930dd_sum(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_vt(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G3),H),T8) = groups7311177749621191930dd_sum(B,A,H,S) ) ) ) ) ) ).

% sum.group
tff(fact_4757_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S: set(B),T8: set(C),G3: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T8))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,G3),S)),T8))
             => ( groups7121269368397514597t_prod(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_vu(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S),G3),H),T8) = groups7121269368397514597t_prod(B,A,H,S) ) ) ) ) ) ).

% prod.group
tff(fact_4758_map__to__set__ran,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),map_to_set(B,A,M)) ).

% map_to_set_ran
tff(fact_4759_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(A,fun(B,B)),A6: set(A),S2: B,T5: B,B5: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( finite4664212375090638736ute_on(A,B,S,G3)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
         => ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A6))
                 => ( aa(A,fun(B,B),F3,X3) = aa(A,fun(B,B),G3,X3) ) )
             => ( ( S2 = T5 )
               => ( ( A6 = B5 )
                 => ( finite_fold(A,B,F3,S2,A6) = finite_fold(A,B,G3,T5,B5) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_4760_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F3),A6)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B5)),aa(set(A),nat,finite_card(A),A6))) ) ) ).

% surj_card_le
tff(fact_4761_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: B,L: list(B)] :
      ( pp(aa(set(A),bool,member(A,aa(B,A,F3,X)),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),L))))
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L)))
          & ( aa(B,A,F3,aa(nat,B,nth(B,L),I)) = aa(B,A,F3,X) ) ) ) ).

% in_set_image_conv_nth
tff(fact_4762_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L4: list(A),F3: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L4) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L)))
           => ( aa(A,B,F3,aa(nat,A,nth(A,L),I3)) = aa(A,B,F3,aa(nat,A,nth(A,L4),I3)) ) )
       => ( aa(set(A),set(B),image2(A,B,F3),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image2(A,B,F3),aa(list(A),set(A),set2(A),L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_4763_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X6: set(A),A6: set(product_prod(A,B)),Y6: set(B),P2: fun(A,fun(B,bool)),Q2: fun(A,fun(B,bool))] :
      ( ( X6 = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A6) )
     => ( ( Y6 = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),A6) )
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => ! [Xa4: B] :
                  ( pp(aa(set(B),bool,member(B,Xa4),Y6))
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X3),Xa4))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q2,X3),Xa4)) ) ) )
         => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2))))
           => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q2)))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_4764_set__to__map__ran,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ran(B,A,set_to_map(B,A,S))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S))) ).

% set_to_map_ran
tff(fact_4765_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(A),set(B),image2(A,B,F3),A6) = finite_fold(A,set(B),aTP_Lamp_vv(fun(A,B),fun(A,fun(set(B),set(B))),F3),bot_bot(set(B)),A6) ) ) ).

% image_fold_insert
tff(fact_4766_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(C),G3: fun(A,B),F3: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite2(C),I5))
         => ( ! [I3: C] :
                ( pp(aa(set(C),bool,member(C,I3),I5))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G3,aa(C,A,F3,I3)))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),groups7311177749621191930dd_sum(A,B,G3,aa(set(C),set(A),image2(C,A,F3),I5))),groups7311177749621191930dd_sum(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,G3),F3),I5))) ) ) ) ).

% sum_image_le
tff(fact_4767_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z4,A6)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_4768_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z4),A6) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_4769_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z4,A6)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z4),A6) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_4770_ran__map__of,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ran(B,A,map_of(B,A,Xs))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)))) ).

% ran_map_of
tff(fact_4771_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),X),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C3),X)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4772_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,C3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
               => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_vw(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
               => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_vw(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3),aa(A,A,aa(A,fun(A,A),times_times(A),X),C3)) ) ) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_vw(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4773_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vx(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vx(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vx(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = 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)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4774_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vy(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vy(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vy(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4775_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vz(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vz(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_vz(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = 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)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4776_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_wa(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_wa(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_wa(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4777_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S: set(A),R2: set(B),G3: fun(A,B),F3: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(set(B),bool,finite_finite2(B),R2))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G3),S)),R2))
             => ( groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_wb(fun(A,B),fun(fun(B,C),fun(A,C)),G3),F3),S) = groups7311177749621191930dd_sum(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_wd(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G3),F3),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4778_set__to__map__insert,axiom,
    ! [B: $tType,A: $tType,Kv: product_prod(A,B),S: set(product_prod(A,B))] :
      ( ~ pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),Kv)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)))
     => ( set_to_map(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert(product_prod(A,B)),Kv),S)) = fun_upd(A,option(B),set_to_map(A,B,S),aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),Kv))) ) ) ).

% set_to_map_insert
tff(fact_4779_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),A6: set(A),X: A,Z4: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( finite_fold(A,B,F3,Z4,A6) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_4780_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_we(fun(A,B),fun(A,product_prod(A,B)),F3)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F3),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_4781_image__mset__map__of,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(multiset(A),multiset(B),image_mset(A,B,aTP_Lamp_wf(list(product_prod(A,B)),fun(A,B),Xs)),mset(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))) = mset(B,aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs)) ) ) ).

% image_mset_map_of
tff(fact_4782_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : subseqs(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),subseqs(A,Xs))),subseqs(A,Xs)) ).

% subseqs.simps(2)
tff(fact_4783_op__conc__empty__img__id,axiom,
    ! [A: $tType,L7: set(list(A))] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A))),L7) = L7 ).

% op_conc_empty_img_id
tff(fact_4784_image__mset__union,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),M6: multiset(B),N7: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,F3),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M6),N7)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(B),multiset(A),image_mset(B,A,F3),M6)),aa(multiset(B),multiset(A),image_mset(B,A,F3),N7)) ).

% image_mset_union
tff(fact_4785_image__map__upd,axiom,
    ! [B: $tType,A: $tType,X: A,A6: set(A),M: fun(A,option(B)),Y: B] :
      ( ~ pp(aa(set(A),bool,member(A,X),A6))
     => ( aa(set(A),set(option(B)),image2(A,option(B),fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y))),A6) = aa(set(A),set(option(B)),image2(A,option(B),M),A6) ) ) ).

% image_map_upd
tff(fact_4786_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D5: set(A),X5: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_bp(A,option(B)),D5),X5) = none(B) ).

% restrict_map_empty
tff(fact_4787_nth__image__indices,axiom,
    ! [A: $tType,L: list(A)] : aa(set(nat),set(A),image2(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_4788_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),D5: set(A),M: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D5))
       => ( restrict_map(A,B,map_upds(A,B,M,Xs,Ys2),D5) = map_upds(A,B,restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(list(A),set(A),set2(A),Xs))),Xs,Ys2) ) ) ) ).

% restrict_map_upds
tff(fact_4789_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_4790_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_wg(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_4791_restrict__map__eq_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A6: set(B),K2: B,V2: A] :
      ( ( aa(B,option(A),restrict_map(B,A,M,A6),K2) = aa(A,option(A),some(A),V2) )
    <=> ( ( aa(B,option(A),M,K2) = aa(A,option(A),some(A),V2) )
        & pp(aa(set(B),bool,member(B,K2),A6)) ) ) ).

% restrict_map_eq(2)
tff(fact_4792_mset__map__split__orig,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),P2: multiset(B),M12: multiset(A),M23: multiset(A)] :
      ( ( aa(multiset(B),multiset(A),image_mset(B,A,F3),P2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M12),M23) )
     => ~ ! [P12: multiset(B),P22: multiset(B)] :
            ( ( P2 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P22) )
           => ( ( aa(multiset(B),multiset(A),image_mset(B,A,F3),P12) = M12 )
             => ( aa(multiset(B),multiset(A),image_mset(B,A,F3),P22) != M23 ) ) ) ) ).

% mset_map_split_orig
tff(fact_4793_mset__map__id,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),G3: fun(A,B),X6: multiset(A)] :
      ( ! [X3: A] : aa(B,A,F3,aa(A,B,G3,X3)) = X3
     => ( aa(multiset(B),multiset(A),image_mset(B,A,F3),aa(multiset(A),multiset(B),image_mset(A,B,G3),X6)) = X6 ) ) ).

% mset_map_id
tff(fact_4794_multiset_Omap__ident,axiom,
    ! [A: $tType,T5: multiset(A)] : aa(multiset(A),multiset(A),image_mset(A,A,aTP_Lamp_cc(A,A)),T5) = T5 ).

% multiset.map_ident
tff(fact_4795_restrict__map__subset__eq,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),R2: set(A),M8: fun(A,option(B)),R8: set(A)] :
      ( ( restrict_map(A,B,M,R2) = M8 )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R8),R2))
       => ( restrict_map(A,B,M,R8) = restrict_map(A,B,M8,R8) ) ) ) ).

% restrict_map_subset_eq
tff(fact_4796_le__map__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M: fun(A,option(B)),X6: set(A)] : pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M,X6)),M)) ) ).

% le_map_restrict
tff(fact_4797_mset__map__split__orig__le,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),P2: multiset(B),M12: multiset(A),M23: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),P2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M12),M23)))
     => ~ ! [P12: multiset(B),P22: multiset(B)] :
            ( ( P2 = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P12),P22) )
           => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),P12)),M12))
             => ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),P22)),M23)) ) ) ) ).

% mset_map_split_orig_le
tff(fact_4798_subset__subseqs,axiom,
    ! [A: $tType,X6: set(A),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(set(set(A)),bool,member(set(A),X6),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))))) ) ).

% subset_subseqs
tff(fact_4799_None__notin__image__Some,axiom,
    ! [A: $tType,A6: set(A)] : ~ pp(aa(set(option(A)),bool,member(option(A),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A6))) ).

% None_notin_image_Some
tff(fact_4800_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2))) ).

% shuffles.simps(3)
tff(fact_4801_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xs: list(A),Ys2: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,Xs,Ys2))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))) ).

% Cons_shuffles_subset2
tff(fact_4802_Cons__shuffles__subset1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys2: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,Ys2))),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2))) ).

% Cons_shuffles_subset1
tff(fact_4803_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A6: set(A),F3: fun(nat,A),N: nat] :
      ( ( A6 = aa(set(nat),set(A),image2(nat,A,F3),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N))) )
     => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ).

% nat_seg_image_imp_finite
tff(fact_4804_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
    <=> ? [N3: nat,F11: fun(nat,A)] : A6 = aa(set(nat),set(A),image2(nat,A,F11),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N3))) ) ).

% finite_conv_nat_seg_image
tff(fact_4805_finite__int__iff__bounded__le,axiom,
    ! [S: set(int)] :
      ( pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S)),aa(int,set(int),set_ord_atMost(int),K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_4806_finite__int__iff__bounded,axiom,
    ! [S: set(int)] :
      ( pp(aa(set(int),bool,finite_finite2(int),S))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S)),aa(int,set(int),set_ord_lessThan(int),K3))) ) ).

% finite_int_iff_bounded
tff(fact_4807_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K2: A,V2: B,M: fun(A,option(B)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V2)),graph(A,B,restrict_map(A,B,M,A6))))
     => pp(aa(set(A),bool,member(A,K2),A6)) ) ).

% graph_restrictD(1)
tff(fact_4808_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y: A,M: fun(B,option(A)),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,Y),ran(B,A,restrict_map(B,A,M,A6))))
     => ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A6))
          & ( aa(B,option(A),M,X3) = aa(A,option(A),some(A),Y) ) ) ) ).

% ran_restrictD
tff(fact_4809_in__image__insert__iff,axiom,
    ! [A: $tType,B5: set(set(A)),X: A,A6: set(A)] :
      ( ! [C8: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),C8),B5))
         => ~ pp(aa(set(A),bool,member(A,X),C8)) )
     => ( pp(aa(set(set(A)),bool,member(set(A),A6),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X)),B5)))
      <=> ( pp(aa(set(A),bool,member(A,X),A6))
          & pp(aa(set(set(A)),bool,member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B5)) ) ) ) ).

% in_image_insert_iff
tff(fact_4810_shuffles_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa2),bot_bot(set(list(A)))) ) )
       => ( ( ( Xa2 = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ! [Y3: A,Ys5: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                   => ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys5))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_4811_map__restrict__insert__none__simp,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),X: B,S2: set(B)] :
      ( ( aa(B,option(A),M,X) = none(A) )
     => ( restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),S2))) = restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),S2)) ) ) ).

% map_restrict_insert_none_simp
tff(fact_4812_restrict__map__upd,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),S: set(A),K2: A,V2: B] : fun_upd(A,option(B),restrict_map(A,B,F3,S),K2,aa(B,option(B),some(B),V2)) = restrict_map(A,B,fun_upd(A,option(B),F3,K2,aa(B,option(B),some(B),V2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K2),S)) ).

% restrict_map_upd
tff(fact_4813_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K2: A,V2: B,M: fun(A,option(B)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V2)),graph(A,B,restrict_map(A,B,M,A6))))
     => ( aa(A,option(B),M,K2) = aa(B,option(B),some(B),V2) ) ) ).

% graph_restrictD(2)
tff(fact_4814_finite__set__image,axiom,
    ! [A: $tType,A6: set(list(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A6)))
     => ( ! [Xs2: list(A)] :
            ( pp(aa(set(list(A)),bool,member(list(A),Xs2),A6))
           => distinct(A,Xs2) )
       => pp(aa(set(list(A)),bool,finite_finite2(list(A)),A6)) ) ) ).

% finite_set_image
tff(fact_4815_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_wh(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_4816_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),set(code_integer),image2(code_integer,code_integer,aTP_Lamp_wi(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_4817_map__upd__eq__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A] : fun_upd(A,option(B),M,X,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ).

% map_upd_eq_restrict
tff(fact_4818_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),U))
     => ( set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U) = aa(set(nat),set(code_integer),image2(nat,code_integer,semiring_1_of_nat(code_integer)),aa(nat,set(nat),set_ord_lessThan(nat),code_nat_of_integer(U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_4819_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_4820_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C3: nat,Y: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C3),Y))
       => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_wj(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,X,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C3)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C3),Y))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_wj(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,X,Y)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_wj(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,X,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4821_shuffles_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_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)),X),Xa2)))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa2),bot_bot(set(list(A)))) )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_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)),Xa2))) ) )
         => ( ( ( Xa2 = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) )
               => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_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)),X),nil(A)))) ) )
           => ~ ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ! [Y3: A,Ys5: list(A)] :
                      ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                     => ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys5))) )
                       => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_4822_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A6: set(A)] : set_Cons(A,A6,aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_wk(A,list(A))),A6) ).

% set_Cons_sing_Nil
tff(fact_4823_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys2: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))))
     => ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys2))) ) ) ).

% shuffles.psimps(3)
tff(fact_4824_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B5: set(A),X: B,A6: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( 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))),image2(A,set(product_prod(B,A)),aTP_Lamp_wl(B,fun(A,set(product_prod(B,A))),X)),B5))),A6) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_vb(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A6,B5) ) ) ).

% fold_union_pair
tff(fact_4825_relInvImage__Id__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B5: set(B)] :
      ( ! [A12: A,A23: A] :
          ( ( aa(A,B,F3,A12) = aa(A,B,F3,A23) )
        <=> ( A12 = A23 ) )
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),bNF_Gr7122648621184425601vImage(A,B,A6,id_on(B,B5),F3)),id2(A))) ) ).

% relInvImage_Id_on
tff(fact_4826_filter__rev__aux__alt,axiom,
    ! [A: $tType,A3: list(A),P2: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P2),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),filter2(A,P2,rev(A,L))),A3) ).

% filter_rev_aux_alt
tff(fact_4827_Sup__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A6: set(fun(A,B)),X: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A6),X) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_wm(A,fun(fun(A,B),B),X)),A6)) ) ).

% Sup_apply
tff(fact_4828_cSup__atMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atMost(A),X)) = X ) ).

% cSup_atMost
tff(fact_4829_cSup__lessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A)
        & no_bot(A) )
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_lessThan(A),X)) = X ) ).

% cSup_lessThan
tff(fact_4830_SUP__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [F3: fun(C,fun(B,A)),A6: set(C),X: 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)),image2(C,fun(B,A),F3),A6)),X) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_wn(fun(C,fun(B,A)),fun(B,fun(C,A)),F3),X)),A6)) ) ).

% SUP_apply
tff(fact_4831_SUP__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [A6: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_wo(A,A)),A6)) = aa(set(A),A,complete_Sup_Sup(A),A6) ) ).

% SUP_identity_eq
tff(fact_4832_UN__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B5: fun(B,set(A)),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
          & pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B5,X4))) ) ) ).

% UN_iff
tff(fact_4833_UN__I,axiom,
    ! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B5,A3)))
       => pp(aa(set(B),bool,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6)))) ) ) ).

% UN_I
tff(fact_4834_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% cSup_singleton
tff(fact_4835_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_4836_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_4837_Sup__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,A6: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A6)) ) ).

% Sup_insert
tff(fact_4838_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,X)) = X ) ) ) ).

% cSup_atLeastLessThan
tff(fact_4839_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_4840_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_4841_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_4842_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_4843_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_4844_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A6: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B5),A6)) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
             => ( aa(B,A,B5,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_4845_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A6: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B5),A6)) = bot_bot(A) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
             => ( aa(B,A,B5,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_4846_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wp(B,A)),A6)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_4847_SUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wq(A,fun(B,A),F3)),A6)) = F3 ) ) ) ).

% SUP_const
tff(fact_4848_cSUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),C3: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wr(A,fun(B,A),C3)),A6)) = C3 ) ) ) ).

% cSUP_const
tff(fact_4849_UN__constant,axiom,
    ! [B: $tType,A: $tType,A6: set(B),C3: set(A)] :
      ( ( ( A6 = bot_bot(set(B)) )
       => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ws(set(A),fun(B,set(A)),C3)),A6)) = bot_bot(set(A)) ) )
      & ( ( A6 != bot_bot(set(B)) )
       => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ws(set(A),fun(B,set(A)),C3)),A6)) = C3 ) ) ) ).

% UN_constant
tff(fact_4850_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ! [A5: A] :
            ( pp(aa(set(A),bool,member(A,A5),A6))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B5,A5))) )
       => pp(aa(set(B),bool,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6)))) ) ) ).

% finite_UN_I
tff(fact_4851_Union__Un__distrib,axiom,
    ! [A: $tType,A6: set(set(A)),B5: set(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))),sup_sup(set(set(A))),A6),B5)) = 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)),A6)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) ).

% Union_Un_distrib
tff(fact_4852_UN__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A6: set(B),B5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = 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)),image2(B,set(A),M6),A6))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),B5))) ).

% UN_Un
tff(fact_4853_UN__singleton,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_wt(A,set(A))),A6)) = A6 ).

% UN_singleton
tff(fact_4854_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C4: set(B),A3: A,B5: fun(B,set(A))] :
      ( ( ( C4 = bot_bot(set(B)) )
       => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wu(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C4)) = bot_bot(set(A)) ) )
      & ( ( C4 != bot_bot(set(B)) )
       => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wu(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C4))) ) ) ) ).

% UN_simps(1)
tff(fact_4855_UN__simps_I3_J,axiom,
    ! [E: $tType,F2: $tType,C4: set(F2),A6: set(E),B5: fun(F2,set(E))] :
      ( ( ( C4 = bot_bot(set(F2)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_wv(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A6),B5)),C4)) = bot_bot(set(E)) ) )
      & ( ( C4 != bot_bot(set(F2)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_wv(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A6),B5)),C4)) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A6),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),B5),C4))) ) ) ) ).

% UN_simps(3)
tff(fact_4856_UN__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C4: set(C),A6: fun(C,set(D)),B5: set(D)] :
      ( ( ( C4 = bot_bot(set(C)) )
       => ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_ww(fun(C,set(D)),fun(set(D),fun(C,set(D))),A6),B5)),C4)) = bot_bot(set(D)) ) )
      & ( ( C4 != bot_bot(set(C)) )
       => ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_ww(fun(C,set(D)),fun(set(D),fun(C,set(D))),A6),B5)),C4)) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A6),C4))),B5) ) ) ) ).

% UN_simps(2)
tff(fact_4857_UN__insert,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A3: B,A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B5,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))) ).

% UN_insert
tff(fact_4858_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)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_4859_UN__extend__simps_I9_J,axiom,
    ! [S8: $tType,R9: $tType,Q8: $tType,C4: fun(R9,set(S8)),B5: fun(Q8,set(R9)),A6: set(Q8)] : aa(set(set(S8)),set(S8),complete_Sup_Sup(set(S8)),aa(set(Q8),set(set(S8)),image2(Q8,set(S8),aa(fun(Q8,set(R9)),fun(Q8,set(S8)),aTP_Lamp_wx(fun(R9,set(S8)),fun(fun(Q8,set(R9)),fun(Q8,set(S8))),C4),B5)),A6)) = aa(set(set(S8)),set(S8),complete_Sup_Sup(set(S8)),aa(set(R9),set(set(S8)),image2(R9,set(S8),C4),aa(set(set(R9)),set(R9),complete_Sup_Sup(set(R9)),aa(set(Q8),set(set(R9)),image2(Q8,set(R9),B5),A6)))) ).

% UN_extend_simps(9)
tff(fact_4860_SUP__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,fun(C,A)),B5: set(C),A6: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aTP_Lamp_wy(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F3),B5)),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(set(B),fun(C,A),aTP_Lamp_xa(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F3),A6)),B5)) ) ).

% SUP_commute
tff(fact_4861_UN__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B5: fun(B,set(A)),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))))
     => ~ ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A6))
           => ~ pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B5,X3))) ) ) ).

% UN_E
tff(fact_4862_SUP__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(C,set(B)),A6: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),G3),A6)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_xb(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F3),G3)),A6)) ) ).

% SUP_UNION
tff(fact_4863_Sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A6: set(fun(A,B)),X5: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A6),X5) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_wm(A,fun(fun(A,B),B),X5)),A6)) ) ).

% Sup_fun_def
tff(fact_4864_UN__UN__flatten,axiom,
    ! [B: $tType,A: $tType,C: $tType,C4: fun(B,set(A)),B5: fun(C,set(B)),A6: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),C4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A6)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_xc(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C4),B5)),A6)) ).

% UN_UN_flatten
tff(fact_4865_Some__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A6)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A6)) ) ) ) ).

% Some_Sup
tff(fact_4866_relcomp__UNION__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S2: set(product_prod(A,C)),R3: fun(D,set(product_prod(C,B))),I5: set(D)] : relcomp(A,C,B,S2,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),image2(D,set(product_prod(C,B)),R3),I5))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_xd(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S2),R3)),I5)) ).

% relcomp_UNION_distrib
tff(fact_4867_relcomp__UNION__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R3: fun(D,set(product_prod(A,C))),I5: set(D),S2: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(set(product_prod(A,C))),set(product_prod(A,C)),complete_Sup_Sup(set(product_prod(A,C))),aa(set(D),set(set(product_prod(A,C))),image2(D,set(product_prod(A,C)),R3),I5)),S2) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_xe(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R3),S2)),I5)) ).

% relcomp_UNION_distrib2
tff(fact_4868_Union__subsetI,axiom,
    ! [A: $tType,A6: set(set(A)),B5: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A6))
         => ? [Y5: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Y5),B5))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Y5)) ) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5))) ) ).

% Union_subsetI
tff(fact_4869_Union__upper,axiom,
    ! [A: $tType,B5: set(A),A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,member(set(A),B5),A6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6))) ) ).

% Union_upper
tff(fact_4870_Union__least,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(A)] :
      ( ! [X10: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X10),A6))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X10),C4)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6)),C4)) ) ).

% Union_least
tff(fact_4871_Union__mono,axiom,
    ! [A: $tType,A6: set(set(A)),B5: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A6),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5))) ) ).

% Union_mono
tff(fact_4872_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A6: set(A),V2: A] :
          ( pp(aa(set(A),bool,member(A,U),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ).

% Sup_upper2
tff(fact_4873_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),B2))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) ) ) ) ).

% Sup_le_iff
tff(fact_4874_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A6: set(A)] :
          ( pp(aa(set(A),bool,member(A,X),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ).

% Sup_upper
tff(fact_4875_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),Z4: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),Z4)) ) ) ).

% Sup_least
tff(fact_4876_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( ! [A5: A] :
              ( pp(aa(set(A),bool,member(A,A5),A6))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),B5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X5)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ).

% Sup_mono
tff(fact_4877_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),X: A] :
          ( ! [Y3: A] :
              ( pp(aa(set(A),bool,member(A,Y3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
         => ( ! [Y3: A] :
                ( ! [Z6: A] :
                    ( pp(aa(set(A),bool,member(A,Z6),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z6),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A6) = X ) ) ) ) ).

% Sup_eqI
tff(fact_4878_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
         => ( ! [Y3: A] :
                ( ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ).

% cSup_eq
tff(fact_4879_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z4: A,X6: set(A)] :
          ( pp(aa(set(A),bool,member(A,Z4),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z4)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = Z4 ) ) ) ) ).

% cSup_eq_maximum
tff(fact_4880_Union__insert,axiom,
    ! [A: $tType,A3: set(A),B5: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),A3),B5)) = 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)),B5)) ).

% Union_insert
tff(fact_4881_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,S: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),S))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ).

% less_Sup_iff
tff(fact_4882_in__Union__o__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: A,Gset: fun(B,set(set(A))),Gmap: fun(C,B),A6: C] :
      ( pp(aa(set(A),bool,member(A,X),aa(C,set(A),aa(fun(C,B),fun(C,set(A)),comp(B,set(A),C,aa(fun(B,set(set(A))),fun(B,set(A)),comp(set(set(A)),set(A),B,complete_Sup_Sup(set(A))),Gset)),Gmap),A6)))
     => pp(aa(set(A),bool,member(A,X),aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A))),aa(fun(C,B),fun(C,set(set(A))),comp(B,set(set(A)),C,Gset),Gmap)),A6))) ) ).

% in_Union_o_assoc
tff(fact_4883_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A6)))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4)) ) ) ) ) ).

% le_Sup_iff
tff(fact_4884_SUP__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B5: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),B5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(C,A,G3,X5))) ) )
         => ( ! [J3: C] :
                ( pp(aa(set(C),bool,member(C,J3),B5))
               => ? [X5: B] :
                    ( pp(aa(set(B),bool,member(B,X5),A6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G3,J3)),aa(B,A,F3,X5))) ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G3),B5)) ) ) ) ) ).

% SUP_eq
tff(fact_4885_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,member(A,V3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V3)) )
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ).

% less_eq_Sup
tff(fact_4886_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z4: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X6)),Z4)) ) ) ) ).

% cSup_least
tff(fact_4887_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
           => ( ! [Y3: A] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),Y3)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_4888_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% le_cSup_finite
tff(fact_4889_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z4: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),aa(set(A),A,complete_Sup_Sup(A),X6)))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X3)) ) ) ) ) ).

% less_cSupD
tff(fact_4890_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
         => ( ( X6 != bot_bot(set(A)) )
           => ~ ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),X6))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X3)) ) ) ) ) ).

% less_cSupE
tff(fact_4891_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),A3)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3)) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_4892_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ).

% Sup_subset_mono
tff(fact_4893_Sup__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,A6: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A6)) ) ).

% Sup_finite_insert
tff(fact_4894_Sup__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ).

% Sup_union_distrib
tff(fact_4895_Union__Int__subset,axiom,
    ! [A: $tType,A6: set(set(A)),B5: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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))),A6),B5))),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)),A6)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)))) ).

% Union_Int_subset
tff(fact_4896_finite__UNION__then__finite,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A6: set(B),A3: B] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))))
     => ( pp(aa(set(B),bool,member(B,A3),A6))
       => pp(aa(set(A),bool,finite_finite2(A),aa(B,set(A),B5,A3))) ) ) ).

% finite_UNION_then_finite
tff(fact_4897_Some__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A6: set(A),F3: fun(A,B)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( aa(B,option(B),some(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A6))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_xf(fun(A,B),fun(A,option(B)),F3)),A6)) ) ) ) ).

% Some_SUP
tff(fact_4898_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),groups7311177749621191930dd_sum(set(A),nat,finite_card(A),U2))) ).

% card_Union_le_sum_card
tff(fact_4899_SUP__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: fun(B,A),X: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),X)) )
         => ( ! [Y3: A] :
                ( ! [I4: B] :
                    ( pp(aa(set(B),bool,member(B,I4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_4900_SUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B5: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( ! [N4: B] :
              ( pp(aa(set(B),bool,member(B,N4),A6))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),B5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N4)),aa(C,A,G3,X5))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G3),B5)))) ) ) ).

% SUP_mono
tff(fact_4901_SUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),U)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),U)) ) ) ).

% SUP_least
tff(fact_4902_SUP__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),A6)))) ) ) ).

% SUP_mono'
tff(fact_4903_SUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I2),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ).

% SUP_upper
tff(fact_4904_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A6: set(B),U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),U))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ).

% SUP_le_iff
tff(fact_4905_SUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),U: A,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I2),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ) ).

% SUP_upper2
tff(fact_4906_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F3: fun(B,A),A6: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,X4))) ) ) ) ).

% less_SUP_iff
tff(fact_4907_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A6: set(B),Y: A,I2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),Y))
         => ( pp(aa(set(B),bool,member(B,I2),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I2)),Y)) ) ) ) ).

% SUP_lessD
tff(fact_4908_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A6: set(B),G3: 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),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),A6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xg(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A6)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_4909_SUP__absorb,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [K2: B,I5: set(B),A6: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,K2),I5))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,A6,K2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,A6),I5))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,A6),I5)) ) ) ) ).

% SUP_absorb
tff(fact_4910_image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),B5: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),image2(B,A,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_xh(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F3),B5)),A6)) ).

% image_UN
tff(fact_4911_UN__extend__simps_I10_J,axiom,
    ! [V4: $tType,U5: $tType,T: $tType,B5: fun(U5,set(V4)),F3: fun(T,U5),A6: set(T)] : aa(set(set(V4)),set(V4),complete_Sup_Sup(set(V4)),aa(set(T),set(set(V4)),image2(T,set(V4),aa(fun(T,U5),fun(T,set(V4)),aTP_Lamp_xi(fun(U5,set(V4)),fun(fun(T,U5),fun(T,set(V4))),B5),F3)),A6)) = aa(set(set(V4)),set(V4),complete_Sup_Sup(set(V4)),aa(set(U5),set(set(V4)),image2(U5,set(V4),B5),aa(set(T),set(U5),image2(T,U5,F3),A6))) ).

% UN_extend_simps(10)
tff(fact_4912_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A)),A6: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6)) = bot_bot(set(A)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
         => ( aa(B,set(A),B5,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_4913_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A)),A6: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
         => ( aa(B,set(A),B5,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_4914_UN__empty,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_4915_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_xj(B,set(A))),A6)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_4916_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,set(A)),G3: fun(C,set(B)),X: C,X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),aa(C,set(B),G3,X)))),X6))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(C,set(B),G3,X)),aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_xk(fun(B,set(A)),fun(set(A),fun(B,bool)),F3),X6)))) ) ).

% UN_image_subset
tff(fact_4917_UN__mono,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(A),F3: fun(A,set(B)),G3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,X3)),aa(A,set(B),G3,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),A6))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G3),B5)))) ) ) ).

% UN_mono
tff(fact_4918_UN__least,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: fun(A,set(B)),C4: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B5,X3)),C4)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6))),C4)) ) ).

% UN_least
tff(fact_4919_UN__upper,axiom,
    ! [B: $tType,A: $tType,A3: A,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B5,A3)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6)))) ) ).

% UN_upper
tff(fact_4920_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A6: fun(B,set(A)),I5: set(B),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))),B5))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),A6,X4)),B5)) ) ) ).

% UN_subset_iff
tff(fact_4921_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A6: set(A),A3: B,B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,U),A6))
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_xl(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B5)),A6)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6))) ) ) ).

% UN_insert_distrib
tff(fact_4922_relInvImage__mono,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),A6: set(B),F3: fun(B,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
     => pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr7122648621184425601vImage(B,A,A6,R1,F3)),bNF_Gr7122648621184425601vImage(B,A,A6,R22,F3))) ) ).

% relInvImage_mono
tff(fact_4923_Int__UN__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A6: fun(B,set(A)),I5: set(B),B5: 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)),image2(B,set(A),A6),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B5),J4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_xn(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A6),B5),J4)),I5)) ).

% Int_UN_distrib2
tff(fact_4924_Int__UN__distrib,axiom,
    ! [A: $tType,B: $tType,B5: set(A),A6: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xo(set(A),fun(fun(B,set(A)),fun(B,set(A))),B5),A6)),I5)) ).

% Int_UN_distrib
tff(fact_4925_UN__extend__simps_I4_J,axiom,
    ! [H10: $tType,G: $tType,A6: fun(G,set(H10)),C4: set(G),B5: set(H10)] : aa(set(H10),set(H10),aa(set(H10),fun(set(H10),set(H10)),inf_inf(set(H10)),aa(set(set(H10)),set(H10),complete_Sup_Sup(set(H10)),aa(set(G),set(set(H10)),image2(G,set(H10),A6),C4))),B5) = aa(set(set(H10)),set(H10),complete_Sup_Sup(set(H10)),aa(set(G),set(set(H10)),image2(G,set(H10),aa(set(H10),fun(G,set(H10)),aTP_Lamp_xp(fun(G,set(H10)),fun(set(H10),fun(G,set(H10))),A6),B5)),C4)) ).

% UN_extend_simps(4)
tff(fact_4926_UN__extend__simps_I5_J,axiom,
    ! [I6: $tType,J5: $tType,A6: set(I6),B5: fun(J5,set(I6)),C4: set(J5)] : aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),inf_inf(set(I6)),A6),aa(set(set(I6)),set(I6),complete_Sup_Sup(set(I6)),aa(set(J5),set(set(I6)),image2(J5,set(I6),B5),C4))) = aa(set(set(I6)),set(I6),complete_Sup_Sup(set(I6)),aa(set(J5),set(set(I6)),image2(J5,set(I6),aa(fun(J5,set(I6)),fun(J5,set(I6)),aTP_Lamp_xq(set(I6),fun(fun(J5,set(I6)),fun(J5,set(I6))),A6),B5)),C4)) ).

% UN_extend_simps(5)
tff(fact_4927_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A6: fun(B,set(A)),B5: fun(B,set(A)),C4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xr(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B5)),C4)) = 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)),image2(B,set(A),A6),C4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C4))) ).

% Un_Union_image
tff(fact_4928_UN__Un__distrib,axiom,
    ! [A: $tType,B: $tType,A6: fun(B,set(A)),B5: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_xr(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B5)),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)),image2(B,set(A),A6),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),I5))) ).

% UN_Un_distrib
tff(fact_4929_UN__absorb,axiom,
    ! [B: $tType,A: $tType,K2: A,I5: set(A),A6: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,K2),I5))
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A6,K2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5)) ) ) ).

% UN_absorb
tff(fact_4930_UN__extend__simps_I6_J,axiom,
    ! [L8: $tType,K6: $tType,A6: fun(K6,set(L8)),C4: set(K6),B5: set(L8)] : aa(set(L8),set(L8),aa(set(L8),fun(set(L8),set(L8)),minus_minus(set(L8)),aa(set(set(L8)),set(L8),complete_Sup_Sup(set(L8)),aa(set(K6),set(set(L8)),image2(K6,set(L8),A6),C4))),B5) = aa(set(set(L8)),set(L8),complete_Sup_Sup(set(L8)),aa(set(K6),set(set(L8)),image2(K6,set(L8),aa(set(L8),fun(K6,set(L8)),aTP_Lamp_xs(fun(K6,set(L8)),fun(set(L8),fun(K6,set(L8))),A6),B5)),C4)) ).

% UN_extend_simps(6)
tff(fact_4931_filter__rev__aux_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,A3: list(A),Xs: list(A)] :
      ( ( pp(aa(A,bool,P2,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A3)),P2),Xs) ) )
      & ( ~ pp(aa(A,bool,P2,X))
       => ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P2),Xs) ) ) ) ).

% filter_rev_aux.simps(2)
tff(fact_4932_filter__rev__aux_Osimps_I1_J,axiom,
    ! [A: $tType,A3: list(A),P2: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P2),nil(A)) = A3 ).

% filter_rev_aux.simps(1)
tff(fact_4933_image__Union,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S: set(set(B))] : aa(set(B),set(A),image2(B,A,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),S)) ).

% image_Union
tff(fact_4934_Union__natural,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : aa(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B)),comp(set(set(B)),set(B),set(set(A)),complete_Sup_Sup(set(B))),image2(set(A),set(B),image2(A,B,F3))) = aa(fun(set(set(A)),set(A)),fun(set(set(A)),set(B)),comp(set(A),set(B),set(set(A)),image2(A,B,F3)),complete_Sup_Sup(set(A))) ).

% Union_natural
tff(fact_4935_Int__Union2,axiom,
    ! [A: $tType,B5: set(set(A)),A6: 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)),B5)),A6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_xt(set(A),fun(set(A),set(A)),A6)),B5)) ).

% Int_Union2
tff(fact_4936_Int__Union,axiom,
    ! [A: $tType,A6: set(A),B5: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6)),B5)) ).

% Int_Union
tff(fact_4937_UN__extend__simps_I8_J,axiom,
    ! [P7: $tType,O: $tType,B5: fun(O,set(P7)),A6: set(set(O))] : aa(set(set(P7)),set(P7),complete_Sup_Sup(set(P7)),aa(set(set(O)),set(set(P7)),image2(set(O),set(P7),aTP_Lamp_xu(fun(O,set(P7)),fun(set(O),set(P7)),B5)),A6)) = aa(set(set(P7)),set(P7),complete_Sup_Sup(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),B5),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A6))) ).

% UN_extend_simps(8)
tff(fact_4938_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F3: fun(B,A),A6: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),aa(B,A,F3,X4))) ) ) ) ) ).

% le_SUP_iff
tff(fact_4939_cSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),M6: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),M6)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),M6)) ) ) ) ).

% cSUP_least
tff(fact_4940_SUP__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),C3: A,F3: fun(B,A)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(B,A,F3,I3))) )
           => ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),I5)) = C3 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),I5))
                 => ( aa(B,A,F3,X4) = C3 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_4941_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_4942_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A6))
                 => ( pp(aa(set(A),bool,member(A,Y3),A6))
                   => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)),A6)) ) )
             => pp(aa(set(A),bool,member(A,aa(set(A),A,complete_Sup_Sup(A),A6)),A6)) ) ) ) ) ).

% finite_Sup_in
tff(fact_4943_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A3)) ) ) ) ).

% cSup_abs_le
tff(fact_4944_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B5: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),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)),A6),B5))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5)))) ) ).

% Sup_inter_less_eq
tff(fact_4945_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B5: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A6)),B5) = finite_fold(A,A,sup_sup(A),B5,A6) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_4946_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),U2))
         => pp(aa(set(A),bool,finite_finite2(A),X3)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),groups7311177749621191930dd_sum(set(A),nat,finite_card(A),U2))) ) ).

% card_Union_le_sum_card_weak
tff(fact_4947_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),bot_bot(set(B))))) = A6 ).

% Union_image_empty
tff(fact_4948_Union__image__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),A3: B,B5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F3,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),B5))) ).

% Union_image_insert
tff(fact_4949_SUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),B5)))) ) ) ) ).

% SUP_subset_mono
tff(fact_4950_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),C3: A] :
          ( ( ( A6 = bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wq(A,fun(B,A),C3)),A6)) = bot_bot(A) ) )
          & ( ( A6 != bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wq(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).

% SUP_constant
tff(fact_4951_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_4952_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A3: B,A6: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A6))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F3,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))) ) ).

% SUP_insert
tff(fact_4953_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A6: set(B),B5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M6),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M6),B5))) ) ).

% SUP_union
tff(fact_4954_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C4: set(B),A3: A,B5: fun(B,set(A))] :
      ( ( ( C4 = bot_bot(set(B)) )
       => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C4))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) ) )
      & ( ( C4 != bot_bot(set(B)) )
       => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_wu(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B5)),C4)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_4955_UN__extend__simps_I3_J,axiom,
    ! [E: $tType,F2: $tType,C4: set(F2),A6: set(E),B5: fun(F2,set(E))] :
      ( ( ( C4 = bot_bot(set(F2)) )
       => ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A6),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),B5),C4))) = A6 ) )
      & ( ( C4 != bot_bot(set(F2)) )
       => ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A6),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),B5),C4))) = aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_wv(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A6),B5)),C4)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_4956_UN__extend__simps_I2_J,axiom,
    ! [D: $tType,C: $tType,C4: set(C),A6: fun(C,set(D)),B5: set(D)] :
      ( ( ( C4 = bot_bot(set(C)) )
       => ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A6),C4))),B5) = B5 ) )
      & ( ( C4 != bot_bot(set(C)) )
       => ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),A6),C4))),B5) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_ww(fun(C,set(D)),fun(set(D),fun(C,set(D))),A6),B5)),C4)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_4957_UN__le__add__shift,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K2: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_xv(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K2)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M6),set_or1337092689740270186AtMost(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)))) ).

% UN_le_add_shift
tff(fact_4958_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K2: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_xv(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K2)),aa(nat,set(nat),set_ord_lessThan(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M6),set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)))) ).

% UN_le_add_shift_strict
tff(fact_4959_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S)),L))),E3)) ) ) ) ).

% cSup_asclose
tff(fact_4960_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( ( S = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = X ) )
            & ( ( S != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,complete_Sup_Sup(A),S)) ) ) ) ) ) ).

% Sup_insert_finite
tff(fact_4961_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,complete_Sup_Sup(A),A6) = finite_fold(A,A,sup_sup(A),bot_bot(A),A6) ) ) ) ).

% Sup_fold_sup
tff(fact_4962_foldl__set,axiom,
    ! [A: $tType,L: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),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),bool),set(set(A)),collect(set(A)),aTP_Lamp_xw(list(set(A)),fun(set(A),bool),L))) ).

% foldl_set
tff(fact_4963_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_xx(fun(B,A),fun(B,set(A)),F3)),A6)) = aa(set(B),set(A),image2(B,A,F3),A6) ).

% UNION_singleton_eq_range
tff(fact_4964_Id__on__def,axiom,
    ! [A: $tType,A6: set(A)] : id_on(A,A6) = 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))),image2(A,set(product_prod(A,A)),aTP_Lamp_xy(A,set(product_prod(A,A)))),A6)) ).

% Id_on_def
tff(fact_4965_sup__SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B5: A,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),B5),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F3),B5,A6) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_4966_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A6: fun(B,set(A)),I2: B,B5: set(A),J4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),A6,I2,B5)),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)),image2(B,set(A),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),bot_bot(set(B))))))),if(set(A),aa(set(B),bool,member(B,I2),J4),B5,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_4967_sum_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),A6: fun(B,set(C)),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A6,X3))) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),I5))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,member(B,Xa4),I5))
                     => ( ( X3 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A6,X3)),aa(B,set(C),A6,Xa4)) = bot_bot(set(C)) ) ) ) )
             => ( groups7311177749621191930dd_sum(C,A,G3,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A6),I5))) = groups7311177749621191930dd_sum(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_xz(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A6),G3),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_4968_prod_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),A6: fun(B,set(C)),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A6,X3))) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),I5))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,member(B,Xa4),I5))
                     => ( ( X3 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A6,X3)),aa(B,set(C),A6,Xa4)) = bot_bot(set(C)) ) ) ) )
             => ( groups7121269368397514597t_prod(C,A,G3,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A6),I5))) = groups7121269368397514597t_prod(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ya(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A6),G3),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_4969_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),I5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),image2(A,set(B),A6),I5)))),groups7311177749621191930dd_sum(A,nat,aTP_Lamp_yb(fun(A,set(B)),fun(A,nat),A6),I5))) ) ).

% card_UN_le
tff(fact_4970_UN__le__eq__Un0,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M6),aa(nat,set(nat),set_ord_atMost(nat),N))) = 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)),image2(nat,set(A),M6),set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M6,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_4971_SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F3),bot_bot(A),A6) ) ) ) ).

% SUP_fold_sup
tff(fact_4972_filter__rev__def,axiom,
    ! [A: $tType] : filter_rev(A) = filter_rev_aux(A,nil(A)) ).

% filter_rev_def
tff(fact_4973_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),I5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),I5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A6,X3))) )
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),I5))
             => ! [Xa4: A] :
                  ( pp(aa(set(A),bool,member(A,Xa4),I5))
                 => ( ( X3 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A6,X3)),aa(A,set(B),A6,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)),image2(A,set(B),A6),I5))) = groups7311177749621191930dd_sum(A,nat,aTP_Lamp_yb(fun(A,set(B)),fun(A,nat),A6),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_4974_finite__subset__Union,axiom,
    ! [A: $tType,A6: set(A),B12: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)))
       => ~ ! [F12: set(set(A))] :
              ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F12))
             => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F12),B12))
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F12))) ) ) ) ) ).

% finite_subset_Union
tff(fact_4975_card__UNION,axiom,
    ! [A: $tType,A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),A6))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),A6))
           => pp(aa(set(A),bool,finite_finite2(A),X3)) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6)) = aa(int,nat,nat2,groups7311177749621191930dd_sum(set(set(A)),int,aTP_Lamp_yc(set(set(A)),int),aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_yd(set(set(A)),fun(set(set(A)),bool),A6)))) ) ) ) ).

% card_UNION
tff(fact_4976_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),A6: set(B),G3: fun(C,A),B5: 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),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G3),B5))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_yf(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F3),G3),B5)),A6)) ) ).

% SUP_inf_distrib2
tff(fact_4977_subset__mset_OcSUP__const,axiom,
    ! [B: $tType,A: $tType,A6: set(B),C3: multiset(A)] :
      ( ( A6 != bot_bot(set(B)) )
     => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aTP_Lamp_yg(multiset(A),fun(B,multiset(A)),C3)),A6)) = C3 ) ) ).

% subset_mset.cSUP_const
tff(fact_4978_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_4979_INF__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [A6: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_yh(A,A)),A6)) = aa(set(A),A,complete_Inf_Inf(A),A6) ) ).

% INF_identity_eq
tff(fact_4980_INT__I,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B2: B,B5: fun(A,set(B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B5,X3))) )
     => pp(aa(set(B),bool,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6)))) ) ).

% INT_I
tff(fact_4981_INT__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B5: fun(B,set(A)),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
         => pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B5,X4))) ) ) ).

% INT_iff
tff(fact_4982_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)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A))))) = none(A) ) ) ).

% singleton_None_Sup
tff(fact_4983_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_4984_Inf__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A6: set(fun(A,B)),X: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A6),X) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_yi(A,fun(fun(A,B),B),X)),A6)) ) ).

% Inf_apply
tff(fact_4985_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A6: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A6) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,member(A,Xa3),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa3),X4)) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_4986_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% cInf_singleton
tff(fact_4987_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,X,Y)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_4988_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_4989_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,X,Y)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_4990_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_4991_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_4992_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_4993_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_4994_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_4995_cINF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),C3: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wr(A,fun(B,A),C3)),A6)) = C3 ) ) ) ).

% cINF_const
tff(fact_4996_INF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wq(A,fun(B,A),F3)),A6)) = F3 ) ) ) ).

% INF_const
tff(fact_4997_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B))] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),I5))
          & pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A6,X5))) )
     => pp(aa(set(B),bool,finite_finite2(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5)))) ) ).

% finite_INT
tff(fact_4998_INF__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [F3: fun(C,fun(B,A)),A6: set(C),X: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Inf_Inf(fun(B,A)),aa(set(C),set(fun(B,A)),image2(C,fun(B,A),F3),A6)),X) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_yj(fun(C,fun(B,A)),fun(B,fun(C,A)),F3),X)),A6)) ) ).

% INF_apply
tff(fact_4999_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A6: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6)) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,member(B,Xa3),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,Xa3)),X4)) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_5000_INT__insert,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A3: B,A6: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B5,A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))) ).

% INT_insert
tff(fact_5001_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A6: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_yk(fun(B,set(A)),fun(B,set(A)),B5)),A6)) ).

% Compl_INT
tff(fact_5002_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A6: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_yk(fun(B,set(A)),fun(B,set(A)),B5)),A6)) ).

% Compl_UN
tff(fact_5003_INF__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,fun(C,A)),B5: set(C),A6: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aTP_Lamp_yl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F3),B5)),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(set(B),fun(C,A),aTP_Lamp_ym(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F3),A6)),B5)) ) ).

% INF_commute
tff(fact_5004_INT__D,axiom,
    ! [A: $tType,B: $tType,B2: A,B5: fun(B,set(A)),A6: set(B),A3: B] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))))
     => ( pp(aa(set(B),bool,member(B,A3),A6))
       => pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B5,A3))) ) ) ).

% INT_D
tff(fact_5005_INT__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B5: fun(B,set(A)),A6: set(B),A3: B] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))))
     => ( ~ pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B5,A3)))
       => ~ pp(aa(set(B),bool,member(B,A3),A6)) ) ) ).

% INT_E
tff(fact_5006_Inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A6: set(fun(A,B)),X5: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A6),X5) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_yi(A,fun(fun(A,B),B),X5)),A6)) ) ).

% Inf_fun_def
tff(fact_5007_Sup__set__def,axiom,
    ! [A: $tType,A6: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_yn(set(set(A)),fun(A,bool),A6)) ).

% Sup_set_def
tff(fact_5008_SUP__Sup__eq,axiom,
    ! [A: $tType,S: set(set(A)),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(set(A)),set(fun(A,bool)),image2(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool))),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S))) ) ).

% SUP_Sup_eq
tff(fact_5009_Some__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] : aa(A,option(A),some(A),aa(set(A),A,complete_Inf_Inf(A),A6)) = aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A6)) ) ).

% Some_Inf
tff(fact_5010_Some__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A6: set(B)] : aa(A,option(A),some(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))) = aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(B),set(option(A)),image2(B,option(A),aTP_Lamp_yo(fun(B,A),fun(B,option(A)),F3)),A6)) ) ).

% Some_INF
tff(fact_5011_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),A6: set(B),G3: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G3),B5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_yq(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F3),G3),B5)),A6)) ) ).

% INF_sup_distrib2
tff(fact_5012_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F3: fun(B,A),B5: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),B5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yr(A,fun(fun(B,A),fun(B,A)),A3),F3)),B5)) ) ).

% sup_INF
tff(fact_5013_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B5)),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_ys(A,fun(A,A),A3)),B5)) ) ).

% Inf_sup
tff(fact_5014_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),B5: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),B5))),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_yt(fun(B,A),fun(A,fun(B,A)),F3),A3)),B5)) ) ).

% INF_sup
tff(fact_5015_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B5: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B5)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),sup_sup(A),A3)),B5)) ) ).

% sup_Inf
tff(fact_5016_Sup__unit__def,axiom,
    ! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Sup_Sup(product_unit),Uu) = product_Unity ).

% Sup_unit_def
tff(fact_5017_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S: set(A),A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S)),A3))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),S))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A3)) ) ) ) ).

% Inf_less_iff
tff(fact_5018_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z4: A,X6: set(A)] :
          ( pp(aa(set(A),bool,member(A,Z4),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = Z4 ) ) ) ) ).

% cInf_eq_minimum
tff(fact_5019_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3)) )
         => ( ! [Y3: A] :
                ( ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X5)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),A3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ).

% cInf_eq
tff(fact_5020_Inter__subset,axiom,
    ! [A: $tType,A6: set(set(A)),B5: set(A)] :
      ( ! [X10: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X10),A6))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X10),B5)) )
     => ( ( A6 != bot_bot(set(set(A))) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6)),B5)) ) ) ).

% Inter_subset
tff(fact_5021_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),X: A] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,member(A,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),I3)) )
         => ( ! [Y3: A] :
                ( ! [I4: A] :
                    ( pp(aa(set(A),bool,member(A,I4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),I4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A6) = X ) ) ) ) ).

% Inf_eqI
tff(fact_5022_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A6: set(A)] :
          ( ! [B4: A] :
              ( pp(aa(set(A),bool,member(A,B4),B5))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B4)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B5))) ) ) ).

% Inf_mono
tff(fact_5023_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A6: set(A)] :
          ( pp(aa(set(A),bool,member(A,X),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),X)) ) ) ).

% Inf_lower
tff(fact_5024_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A6: set(A),V2: A] :
          ( pp(aa(set(A),bool,member(A,U),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),V2)) ) ) ) ).

% Inf_lower2
tff(fact_5025_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A6)))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X4)) ) ) ) ).

% le_Inf_iff
tff(fact_5026_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),Z4: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),aa(set(A),A,complete_Inf_Inf(A),A6))) ) ) ).

% Inf_greatest
tff(fact_5027_Inter__lower,axiom,
    ! [A: $tType,B5: set(A),A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,member(set(A),B5),A6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6)),B5)) ) ).

% Inter_lower
tff(fact_5028_Inter__greatest,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(A)] :
      ( ! [X10: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X10),A6))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),X10)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6))) ) ).

% Inter_greatest
tff(fact_5029_Inter__anti__mono,axiom,
    ! [A: $tType,B5: set(set(A)),A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B5),A6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B5))) ) ).

% Inter_anti_mono
tff(fact_5030_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),X))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) ) ) ) ) ).

% Inf_le_iff
tff(fact_5031_INF__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B5: set(C),G3: fun(C,A),F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),B5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G3,X5)),aa(B,A,F3,I3))) ) )
         => ( ! [J3: C] :
                ( pp(aa(set(C),bool,member(C,J3),B5))
               => ? [X5: B] :
                    ( pp(aa(set(B),bool,member(B,X5),A6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X5)),aa(C,A,G3,J3))) ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G3),B5)) ) ) ) ) ).

% INF_eq
tff(fact_5032_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3)) )
           => ( ! [Y3: A] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X5)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),A3)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_5033_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z4: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ).

% cInf_greatest
tff(fact_5034_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,member(A,V3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),U)) )
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),U)) ) ) ) ).

% Inf_less_eq
tff(fact_5035_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_le_finite
tff(fact_5036_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z4: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z4))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z4)) ) ) ) ) ).

% cInf_lessD
tff(fact_5037_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X3)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_5038_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B5))) ) ) ).

% Inf_superset_mono
tff(fact_5039_Inf__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ).

% Inf_union_distrib
tff(fact_5040_Inter__Un__subset,axiom,
    ! [A: $tType,A6: set(set(A)),B5: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(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_Inf_Inf(set(A)),A6)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B5))),aa(set(set(A)),set(A),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))),A6),B5)))) ).

% Inter_Un_subset
tff(fact_5041_INF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),U: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ).

% INF_greatest
tff(fact_5042_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F3: fun(B,A),A6: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ).

% le_INF_iff
tff(fact_5043_INF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),F3: fun(B,A),U: A] :
          ( pp(aa(set(B),bool,member(B,I2),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),U)) ) ) ) ).

% INF_lower2
tff(fact_5044_INF__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),A6)))) ) ) ).

% INF_mono'
tff(fact_5045_INF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I2),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(B,A,F3,I2))) ) ) ).

% INF_lower
tff(fact_5046_INF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(B),A6: set(C),F3: fun(C,A),G3: fun(B,A)] :
          ( ! [M5: B] :
              ( pp(aa(set(B),bool,member(B,M5),B5))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,member(C,X5),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X5)),aa(B,A,G3,M5))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),B5)))) ) ) ).

% INF_mono
tff(fact_5047_INF__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),X: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F3,I3))) )
         => ( ! [Y3: A] :
                ( ! [I4: B] :
                    ( pp(aa(set(B),bool,member(B,I4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),aa(B,A,F3,I4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6)) = X ) ) ) ) ).

% INF_eqI
tff(fact_5048_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F3: fun(B,A),A6: set(B),I2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))))
         => ( pp(aa(set(B),bool,member(B,I2),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F3,I2))) ) ) ) ).

% less_INF_D
tff(fact_5049_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A6: set(B),A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),A3))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),A3)) ) ) ) ).

% INF_less_iff
tff(fact_5050_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,set(A)),S: set(B),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),aTP_Lamp_yu(fun(B,set(A)),fun(B,fun(A,bool)),R3)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R3),S)))) ) ).

% SUP_UN_eq
tff(fact_5051_INF__absorb,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [K2: B,I5: set(B),A6: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,K2),I5))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,A6,K2)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,A6),I5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,A6),I5)) ) ) ) ).

% INF_absorb
tff(fact_5052_INF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A6: set(B),G3: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),A6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_yv(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A6)) ) ).

% INF_inf_distrib
tff(fact_5053_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool)))),S)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S))) ) ).

% SUP_Sup_eq2
tff(fact_5054_INT__extend__simps_I10_J,axiom,
    ! [V4: $tType,U5: $tType,T: $tType,B5: fun(U5,set(V4)),F3: fun(T,U5),A6: set(T)] : aa(set(set(V4)),set(V4),complete_Inf_Inf(set(V4)),aa(set(T),set(set(V4)),image2(T,set(V4),aa(fun(T,U5),fun(T,set(V4)),aTP_Lamp_xi(fun(U5,set(V4)),fun(fun(T,U5),fun(T,set(V4))),B5),F3)),A6)) = aa(set(set(V4)),set(V4),complete_Inf_Inf(set(V4)),aa(set(U5),set(set(V4)),image2(U5,set(V4),B5),aa(set(T),set(U5),image2(T,U5,F3),A6))) ).

% INT_extend_simps(10)
tff(fact_5055_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B5: set(A),A6: fun(B,set(A)),I5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(B,set(A),A6,X4))) ) ) ).

% INT_subset_iff
tff(fact_5056_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(A),F3: fun(A,set(B)),G3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,X3)),aa(A,set(B),G3,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),B5))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G3),A6)))) ) ) ).

% INT_anti_mono
tff(fact_5057_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A6: set(A),C4: set(B),B5: fun(A,set(B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),aa(A,set(B),B5,X3))) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6)))) ) ).

% INT_greatest
tff(fact_5058_INT__lower,axiom,
    ! [B: $tType,A: $tType,A3: A,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6))),aa(A,set(B),B5,A3))) ) ).

% INT_lower
tff(fact_5059_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A6: set(A),A3: B,B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,U),A6))
     => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_xl(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B5)),A6)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A6))) ) ) ).

% INT_insert_distrib
tff(fact_5060_INT__extend__simps_I5_J,axiom,
    ! [I6: $tType,J5: $tType,A3: I6,B5: fun(J5,set(I6)),C4: set(J5)] : aa(set(I6),set(I6),aa(I6,fun(set(I6),set(I6)),insert(I6),A3),aa(set(set(I6)),set(I6),complete_Inf_Inf(set(I6)),aa(set(J5),set(set(I6)),image2(J5,set(I6),B5),C4))) = aa(set(set(I6)),set(I6),complete_Inf_Inf(set(I6)),aa(set(J5),set(set(I6)),image2(J5,set(I6),aa(fun(J5,set(I6)),fun(J5,set(I6)),aTP_Lamp_yw(I6,fun(fun(J5,set(I6)),fun(J5,set(I6))),A3),B5)),C4)) ).

% INT_extend_simps(5)
tff(fact_5061_Inter__Un__distrib,axiom,
    ! [A: $tType,A6: set(set(A)),B5: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(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))),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B5)) ).

% Inter_Un_distrib
tff(fact_5062_INT__absorb,axiom,
    ! [B: $tType,A: $tType,K2: A,I5: set(A),A6: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,K2),I5))
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A6,K2)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5)) ) ) ).

% INT_absorb
tff(fact_5063_INT__Int__distrib,axiom,
    ! [A: $tType,B: $tType,A6: fun(B,set(A)),B5: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_yx(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B5)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),I5))) ).

% INT_Int_distrib
tff(fact_5064_Int__Inter__image,axiom,
    ! [A: $tType,B: $tType,A6: fun(B,set(A)),B5: fun(B,set(A)),C4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_yx(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B5)),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),C4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C4))) ).

% Int_Inter_image
tff(fact_5065_INT__extend__simps_I7_J,axiom,
    ! [M10: $tType,N9: $tType,A6: set(M10),B5: fun(N9,set(M10)),C4: set(N9)] : aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),sup_sup(set(M10)),A6),aa(set(set(M10)),set(M10),complete_Inf_Inf(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),B5),C4))) = aa(set(set(M10)),set(M10),complete_Inf_Inf(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_yy(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),A6),B5)),C4)) ).

% INT_extend_simps(7)
tff(fact_5066_INT__extend__simps_I6_J,axiom,
    ! [L8: $tType,K6: $tType,A6: fun(K6,set(L8)),C4: set(K6),B5: set(L8)] : aa(set(L8),set(L8),aa(set(L8),fun(set(L8),set(L8)),sup_sup(set(L8)),aa(set(set(L8)),set(L8),complete_Inf_Inf(set(L8)),aa(set(K6),set(set(L8)),image2(K6,set(L8),A6),C4))),B5) = aa(set(set(L8)),set(L8),complete_Inf_Inf(set(L8)),aa(set(K6),set(set(L8)),image2(K6,set(L8),aa(set(L8),fun(K6,set(L8)),aTP_Lamp_yz(fun(K6,set(L8)),fun(set(L8),fun(K6,set(L8))),A6),B5)),C4)) ).

% INT_extend_simps(6)
tff(fact_5067_Un__INT__distrib,axiom,
    ! [A: $tType,B: $tType,B5: set(A),A6: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_za(set(A),fun(fun(B,set(A)),fun(B,set(A))),B5),A6)),I5)) ).

% Un_INT_distrib
tff(fact_5068_Un__INT__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A6: fun(B,set(A)),I5: set(B),B5: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B5),J4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_zc(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A6),B5),J4)),I5)) ).

% Un_INT_distrib2
tff(fact_5069_INT__extend__simps_I9_J,axiom,
    ! [S8: $tType,R9: $tType,Q8: $tType,C4: fun(R9,set(S8)),B5: fun(Q8,set(R9)),A6: set(Q8)] : aa(set(set(S8)),set(S8),complete_Inf_Inf(set(S8)),aa(set(Q8),set(set(S8)),image2(Q8,set(S8),aa(fun(Q8,set(R9)),fun(Q8,set(S8)),aTP_Lamp_zd(fun(R9,set(S8)),fun(fun(Q8,set(R9)),fun(Q8,set(S8))),C4),B5)),A6)) = aa(set(set(S8)),set(S8),complete_Inf_Inf(set(S8)),aa(set(R9),set(set(S8)),image2(R9,set(S8),C4),aa(set(set(R9)),set(R9),complete_Sup_Sup(set(R9)),aa(set(Q8),set(set(R9)),image2(Q8,set(R9),B5),A6)))) ).

% INT_extend_simps(9)
tff(fact_5070_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B12: set(set(A)),A6: set(A)] :
      ( ( ( B12 = bot_bot(set(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_Inf_Inf(set(A)),B12)),A6) = A6 ) )
      & ( ( B12 != bot_bot(set(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_Inf_Inf(set(A)),B12)),A6) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_xt(set(A),fun(set(A),set(A)),A6)),B12)) ) ) ) ).

% Int_Inter_eq(2)
tff(fact_5071_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,B12: set(set(A)),A6: set(A)] :
      ( ( ( B12 = bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)) = A6 ) )
      & ( ( B12 != bot_bot(set(set(A))) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6)),B12)) ) ) ) ).

% Int_Inter_eq(1)
tff(fact_5072_Un__Inter,axiom,
    ! [A: $tType,A6: set(A),B5: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B5)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6)),B5)) ).

% Un_Inter
tff(fact_5073_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A6: set(B),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),X))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y4))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),Y4)) ) ) ) ) ).

% INF_le_iff
tff(fact_5074_INF__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),C3: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),C3)) )
           => ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),I5)) = C3 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),I5))
                 => ( aa(B,A,F3,X4) = C3 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_5075_cINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),M: A,F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ) ).

% cINF_greatest
tff(fact_5076_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X4)) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_5077_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ).

% Inf_le_Sup
tff(fact_5078_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S))),A3)) ) ) ) ).

% cInf_abs_ge
tff(fact_5079_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B5: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))) ) ).

% less_eq_Inf_inter
tff(fact_5080_INF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(B),A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),B5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),B5)))) ) ) ) ).

% INF_superset_mono
tff(fact_5081_INF__inf__const2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),X: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_ze(fun(B,A),fun(A,fun(B,A)),F3),X)),I5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),I5))),X) ) ) ) ).

% INF_inf_const2
tff(fact_5082_INF__inf__const1,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),X: A,F3: fun(B,A)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zf(A,fun(fun(B,A),fun(B,A)),X),F3)),I5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_5083_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B5: fun(B,A),A6: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B5),A6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_zg(fun(B,A),fun(B,A),B5)),A6)) ) ).

% uminus_SUP
tff(fact_5084_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B5: fun(B,A),A6: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B5),A6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_zg(fun(B,A),fun(B,A),B5)),A6)) ) ).

% uminus_INF
tff(fact_5085_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A3: B,A6: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A6))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F3,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))) ) ).

% INF_insert
tff(fact_5086_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A6: set(B),B5: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M6),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M6),B5))) ) ).

% INF_union
tff(fact_5087_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,bool))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),S),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),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),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S))))) ) ).

% Sup_SUP_eq2
tff(fact_5088_INT__extend__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C4: set(D),A6: set(C),B5: fun(D,set(C))] :
      ( ( ( C4 = bot_bot(set(D)) )
       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A6),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B5),C4))) = A6 ) )
      & ( ( C4 != bot_bot(set(D)) )
       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A6),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B5),C4))) = aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_zh(set(C),fun(fun(D,set(C)),fun(D,set(C))),A6),B5)),C4)) ) ) ) ).

% INT_extend_simps(2)
tff(fact_5089_INT__extend__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,C4: set(A),A6: fun(A,set(B)),B5: set(B)] :
      ( ( ( C4 = bot_bot(set(A)) )
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),C4))),B5) = B5 ) )
      & ( ( C4 != bot_bot(set(A)) )
       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),C4))),B5) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_zi(fun(A,set(B)),fun(set(B),fun(A,set(B))),A6),B5)),C4)) ) ) ) ).

% INT_extend_simps(1)
tff(fact_5090_INT__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A6: set(B),B5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),A6))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M6),B5))) ).

% INT_Un
tff(fact_5091_UN__extend__simps_I7_J,axiom,
    ! [M10: $tType,N9: $tType,A6: set(M10),B5: fun(N9,set(M10)),C4: set(N9)] : aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),minus_minus(set(M10)),A6),aa(set(set(M10)),set(M10),complete_Inf_Inf(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),B5),C4))) = aa(set(set(M10)),set(M10),complete_Sup_Sup(set(M10)),aa(set(N9),set(set(M10)),image2(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_zj(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),A6),B5)),C4)) ).

% UN_extend_simps(7)
tff(fact_5092_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(C,set(product_prod(A,B))),S: set(C),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_zk(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R3)),S)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),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))),image2(C,set(product_prod(A,B)),R3),S)))) ) ).

% SUP_UN_eq2
tff(fact_5093_INT__extend__simps_I8_J,axiom,
    ! [P7: $tType,O: $tType,B5: fun(O,set(P7)),A6: set(set(O))] : aa(set(set(P7)),set(P7),complete_Inf_Inf(set(P7)),aa(set(set(O)),set(set(P7)),image2(set(O),set(P7),aTP_Lamp_zl(fun(O,set(P7)),fun(set(O),set(P7)),B5)),A6)) = aa(set(set(P7)),set(P7),complete_Inf_Inf(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),B5),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A6))) ).

% INT_extend_simps(8)
tff(fact_5094_INF__le__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ).

% INF_le_SUP
tff(fact_5095_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S)),L))),E3)) ) ) ) ).

% cInf_asclose
tff(fact_5096_INT__extend__simps_I4_J,axiom,
    ! [G: $tType,H10: $tType,C4: set(H10),A6: set(G),B5: fun(H10,set(G))] :
      ( ( ( C4 = bot_bot(set(H10)) )
       => ( aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),A6),aa(set(set(G)),set(G),complete_Sup_Sup(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),B5),C4))) = A6 ) )
      & ( ( C4 != bot_bot(set(H10)) )
       => ( aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),A6),aa(set(set(G)),set(G),complete_Sup_Sup(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),B5),C4))) = aa(set(set(G)),set(G),complete_Inf_Inf(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_zm(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),A6),B5)),C4)) ) ) ) ).

% INT_extend_simps(4)
tff(fact_5097_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),B5: set(B),A3: 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),image2(B,A,F3),B5))),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_zn(fun(B,A),fun(A,fun(B,A)),F3),A3)),B5)) ) ).

% SUP_inf
tff(fact_5098_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_zo(A,fun(A,A),A3)),B5)) ) ).

% Sup_inf
tff(fact_5099_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F3: fun(B,A),B5: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),B5))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zp(A,fun(fun(B,A),fun(B,A)),A3),F3)),B5)) ) ).

% inf_SUP
tff(fact_5100_UN__UN__split__split__eq,axiom,
    ! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,A6: fun(B,fun(C,fun(D,fun(E,set(A))))),Y6: set(product_prod(D,E)),X6: 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)),image2(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(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_zq(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),A6),Y6))),X6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image2(product_prod(B,C),set(A),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_zt(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),A6),Y6)),X6)) ).

% UN_UN_split_split_eq
tff(fact_5101_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ).

% length_remdups_concat
tff(fact_5102_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F3: fun(nat,set(A)),S: set(A)] :
      ( ! [I3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F3,I3)),S))
     => ( pp(aa(set(A),bool,finite_finite2(A),S))
       => ( ? [N10: nat] :
              ( ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N10))
                 => ! [M5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N10))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N4))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F3,M5)),aa(nat,set(A),F3,N4))) ) ) )
              & ! [N4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N10),N4))
                 => ( aa(nat,set(A),F3,N10) = aa(nat,set(A),F3,N4) ) ) )
         => ( aa(nat,set(A),F3,aa(set(A),nat,finite_card(A),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F3),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5103_top__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( top(C)
     => ! [X: D] : aa(D,C,top_top(fun(D,C)),X) = top_top(C) ) ).

% top_apply
tff(fact_5104_UNIV__I,axiom,
    ! [A: $tType,X: A] : pp(aa(set(A),bool,member(A,X),top_top(set(A)))) ).

% UNIV_I
tff(fact_5105_INF2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A6: set(A),B5: fun(A,fun(B,fun(C,bool))),B2: B,C3: C] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),B5,X3),B2),C3)) )
     => pp(aa(C,bool,aa(B,fun(C,bool),aa(set(fun(B,fun(C,bool))),fun(B,fun(C,bool)),complete_Inf_Inf(fun(B,fun(C,bool))),aa(set(A),set(fun(B,fun(C,bool))),image2(A,fun(B,fun(C,bool)),B5),A6)),B2),C3)) ) ).

% INF2_I
tff(fact_5106_INF1__I,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: fun(A,fun(B,bool)),B2: B] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => pp(aa(B,bool,aa(A,fun(B,bool),B5,X3),B2)) )
     => pp(aa(B,bool,aa(set(fun(B,bool)),fun(B,bool),complete_Inf_Inf(fun(B,bool)),aa(set(A),set(fun(B,bool)),image2(A,fun(B,bool),B5),A6)),B2)) ) ).

% INF1_I
tff(fact_5107_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),top_top(set(option(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_option_UNIV
tff(fact_5108_boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% boolean_algebra.disj_one_right
tff(fact_5109_boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% boolean_algebra.disj_one_left
tff(fact_5110_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_5111_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% sup_top_left
tff(fact_5112_Int__UNIV,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = top_top(set(A)) )
    <=> ( ( A6 = top_top(set(A)) )
        & ( B5 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_5113_card__eq__UNIV2,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) = aa(set(A),nat,finite_card(A),S) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV2
tff(fact_5114_card__eq__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),S) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_eq_UNIV
tff(fact_5115_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).

% max_top
tff(fact_5116_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_5117_restrict__map__UNIV,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B))] : restrict_map(A,B,F3,top_top(set(A))) = F3 ).

% restrict_map_UNIV
tff(fact_5118_Collect__const,axiom,
    ! [A: $tType,P2: bool] :
      ( ( pp(P2)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_zu(bool,fun(A,bool),P2)) = top_top(set(A)) ) )
      & ( ~ pp(P2)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_zu(bool,fun(A,bool),P2)) = bot_bot(set(A)) ) ) ) ).

% Collect_const
tff(fact_5119_finite__Collect__not,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2))))
      <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_Collect_not
tff(fact_5120_subset__mset_OcINF__const,axiom,
    ! [B: $tType,A: $tType,A6: set(B),C3: multiset(A)] :
      ( ( A6 != bot_bot(set(B)) )
     => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aTP_Lamp_yg(multiset(A),fun(B,multiset(A)),C3)),A6)) = C3 ) ) ).

% subset_mset.cINF_const
tff(fact_5121_None__in__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(option(A))] :
          ( pp(aa(set(option(A)),bool,member(option(A),none(A)),A6))
         => ( aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),A6) = none(A) ) ) ) ).

% None_in_Inf
tff(fact_5122_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_5123_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A6: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A6) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,member(A,Xa3),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Xa3)) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_5124_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_5125_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_5126_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y)) = top_top(A) ) ).

% sup_compl_top_left2
tff(fact_5127_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = top_top(A) ) ).

% sup_compl_top_left1
tff(fact_5128_SUP2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A3: A,A6: set(A),B5: fun(A,fun(B,fun(C,bool))),B2: B,C3: C] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),B5,A3),B2),C3))
       => pp(aa(C,bool,aa(B,fun(C,bool),aa(set(fun(B,fun(C,bool))),fun(B,fun(C,bool)),complete_Sup_Sup(fun(B,fun(C,bool))),aa(set(A),set(fun(B,fun(C,bool))),image2(A,fun(B,fun(C,bool)),B5),A6)),B2),C3)) ) ) ).

% SUP2_I
tff(fact_5129_Diff__UNIV,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_5130_card__ge__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(A),nat,finite_card(A),S)))
        <=> ( S = top_top(set(A)) ) ) ) ).

% card_ge_UNIV
tff(fact_5131_SUP1__I,axiom,
    ! [A: $tType,B: $tType,A3: A,A6: set(A),B5: fun(A,fun(B,bool)),B2: B] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),B5,A3),B2))
       => pp(aa(B,bool,aa(set(fun(B,bool)),fun(B,bool),complete_Sup_Sup(fun(B,bool)),aa(set(A),set(fun(B,bool)),image2(A,fun(B,bool),B5),A6)),B2)) ) ) ).

% SUP1_I
tff(fact_5132_surj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( ( aa(set(A),set(A),image2(A,A,F3),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image2(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_5133_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_remdups_leq
tff(fact_5134_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_vo(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_5135_INF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_zv(B,A)),A6)) = top_top(A) ) ).

% INF_top
tff(fact_5136_INF__top__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A6: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B5),A6)) = top_top(A) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
             => ( aa(B,A,B5,X4) = top_top(A) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_5137_INF__top__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A6: set(B)] :
          ( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,B5),A6)) )
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A6))
             => ( aa(B,A,B5,X4) = top_top(A) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_5138_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A6: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,member(B,Xa3),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),aa(B,A,F3,Xa3))) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_5139_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_or(A,fun(B,A)),X)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ).

% range_constant
tff(fact_5140_INT__constant,axiom,
    ! [B: $tType,A: $tType,A6: set(B),C3: set(A)] :
      ( ( ( A6 = bot_bot(set(B)) )
       => ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ws(set(A),fun(B,set(A)),C3)),A6)) = top_top(set(A)) ) )
      & ( ( A6 != bot_bot(set(B)) )
       => ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ws(set(A),fun(B,set(A)),C3)),A6)) = C3 ) ) ) ).

% INT_constant
tff(fact_5141_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),X)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_5142_INT__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C4: set(A),A6: fun(A,set(B)),B5: set(B)] :
      ( ( ( C4 = bot_bot(set(A)) )
       => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_zi(fun(A,set(B)),fun(set(B),fun(A,set(B))),A6),B5)),C4)) = top_top(set(B)) ) )
      & ( ( C4 != bot_bot(set(A)) )
       => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_zi(fun(A,set(B)),fun(set(B),fun(A,set(B))),A6),B5)),C4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),C4))),B5) ) ) ) ).

% INT_simps(1)
tff(fact_5143_INT__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C4: set(D),A6: set(C),B5: fun(D,set(C))] :
      ( ( ( C4 = bot_bot(set(D)) )
       => ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_zh(set(C),fun(fun(D,set(C)),fun(D,set(C))),A6),B5)),C4)) = top_top(set(C)) ) )
      & ( ( C4 != bot_bot(set(D)) )
       => ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_zh(set(C),fun(fun(D,set(C)),fun(D,set(C))),A6),B5)),C4)) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A6),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),image2(D,set(C),B5),C4))) ) ) ) ).

% INT_simps(2)
tff(fact_5144_INT__simps_I3_J,axiom,
    ! [E: $tType,F2: $tType,C4: set(E),A6: fun(E,set(F2)),B5: set(F2)] :
      ( ( ( C4 = bot_bot(set(E)) )
       => ( aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_zw(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A6),B5)),C4)) = top_top(set(F2)) ) )
      & ( ( C4 != bot_bot(set(E)) )
       => ( aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_zw(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A6),B5)),C4)) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),A6),C4))),B5) ) ) ) ).

% INT_simps(3)
tff(fact_5145_INT__simps_I4_J,axiom,
    ! [G: $tType,H10: $tType,C4: set(H10),A6: set(G),B5: fun(H10,set(G))] :
      ( ( ( C4 = bot_bot(set(H10)) )
       => ( aa(set(set(G)),set(G),complete_Inf_Inf(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_zm(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),A6),B5)),C4)) = top_top(set(G)) ) )
      & ( ( C4 != bot_bot(set(H10)) )
       => ( aa(set(set(G)),set(G),complete_Inf_Inf(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_zm(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),A6),B5)),C4)) = aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),A6),aa(set(set(G)),set(G),complete_Sup_Sup(set(G)),aa(set(H10),set(set(G)),image2(H10,set(G),B5),C4))) ) ) ) ).

% INT_simps(4)
tff(fact_5146_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool)))),S)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S))) ) ).

% INF_Int_eq2
tff(fact_5147_INF1__D,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,fun(A,bool)),A6: set(B),B2: A,A3: B] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),B5),A6)),B2))
     => ( pp(aa(set(B),bool,member(B,A3),A6))
       => pp(aa(A,bool,aa(B,fun(A,bool),B5,A3),B2)) ) ) ).

% INF1_D
tff(fact_5148_INF1__E,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,fun(A,bool)),A6: set(B),B2: A,A3: B] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),B5),A6)),B2))
     => ( ~ pp(aa(A,bool,aa(B,fun(A,bool),B5,A3),B2))
       => ~ pp(aa(set(B),bool,member(B,A3),A6)) ) ) ).

% INF1_E
tff(fact_5149_INF2__D,axiom,
    ! [A: $tType,C: $tType,B: $tType,B5: fun(C,fun(A,fun(B,bool))),A6: set(C),B2: A,C3: B,A3: C] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),B5),A6)),B2),C3))
     => ( pp(aa(set(C),bool,member(C,A3),A6))
       => pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B5,A3),B2),C3)) ) ) ).

% INF2_D
tff(fact_5150_INF2__E,axiom,
    ! [B: $tType,A: $tType,C: $tType,B5: fun(C,fun(A,fun(B,bool))),A6: set(C),B2: A,C3: B,A3: C] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),B5),A6)),B2),C3))
     => ( ~ pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B5,A3),B2),C3))
       => ~ pp(aa(set(C),bool,member(C,A3),A6)) ) ) ).

% INF2_E
tff(fact_5151_SUP1__E,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,fun(A,bool)),A6: set(B),B2: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),B5),A6)),B2))
     => ~ ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A6))
           => ~ pp(aa(A,bool,aa(B,fun(A,bool),B5,X3),B2)) ) ) ).

% SUP1_E
tff(fact_5152_SUP2__E,axiom,
    ! [A: $tType,C: $tType,B: $tType,B5: fun(C,fun(A,fun(B,bool))),A6: set(C),B2: A,C3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),B5),A6)),B2),C3))
     => ~ ! [X3: C] :
            ( pp(aa(set(C),bool,member(C,X3),A6))
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B5,X3),B2),C3)) ) ) ).

% SUP2_E
tff(fact_5153_Inf__set__def,axiom,
    ! [A: $tType,A6: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_zx(set(set(A)),fun(A,bool),A6)) ).

% Inf_set_def
tff(fact_5154_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))))
     => ~ ! [X3: B] : B2 != aa(B,A,F3,X3) ) ).

% rangeE
tff(fact_5155_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),G3: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_zy(fun(C,A),fun(fun(B,C),fun(B,A)),F3),G3)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F3),aa(set(B),set(C),image2(B,C,G3),top_top(set(B)))) ).

% range_composition
tff(fact_5156_INF__Int__eq,axiom,
    ! [A: $tType,S: set(set(A)),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(set(A)),set(fun(A,bool)),image2(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool))),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S))) ) ).

% INF_Int_eq
tff(fact_5157_rangeI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: B] : pp(aa(set(A),bool,member(A,aa(B,A,F3,X)),aa(set(B),set(A),image2(B,A,F3),top_top(set(B))))) ).

% rangeI
tff(fact_5158_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),X: B] :
      ( ( B2 = aa(B,A,F3,X) )
     => pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F3),top_top(set(B))))) ) ).

% range_eqI
tff(fact_5159_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_5160_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_5161_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_5162_Inf__unit__def,axiom,
    ! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Inf_Inf(product_unit),Uu) = product_Unity ).

% Inf_unit_def
tff(fact_5163_Int__UNIV__right,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),top_top(set(A))) = A6 ).

% Int_UNIV_right
tff(fact_5164_Int__UNIV__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B5) = B5 ).

% Int_UNIV_left
tff(fact_5165_Inf__nat__def1,axiom,
    ! [K5: set(nat)] :
      ( ( K5 != bot_bot(set(nat)) )
     => pp(aa(set(nat),bool,member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K5)),K5)) ) ).

% Inf_nat_def1
tff(fact_5166_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_5167_insert__UNIV,axiom,
    ! [A: $tType,X: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_5168_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),top_top(A))) ) ).

% top_greatest
tff(fact_5169_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_5170_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5171_subset__UNIV,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),top_top(set(A)))) ).

% subset_UNIV
tff(fact_5172_Un__UNIV__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B5) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_5173_Un__UNIV__right,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_5174_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_5175_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),top_top(A))) ) ) ).

% top.not_eq_extremum
tff(fact_5176_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A3)) ) ).

% top.extremum_strict
tff(fact_5177_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X3: A] : pp(aa(set(A),bool,member(A,X3),top_top(set(A)))) ).

% UNIV_witness
tff(fact_5178_UNIV__eq__I,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ! [X3: A] : pp(aa(set(A),bool,member(A,X3),A6))
     => ( top_top(set(A)) = A6 ) ) ).

% UNIV_eq_I
tff(fact_5179_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_av(A,bool)) ).

% UNIV_def
tff(fact_5180_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B2)) = top_top(A) ) ).

% sup_cancel_left2
tff(fact_5181_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B2)) = top_top(A) ) ).

% sup_cancel_left1
tff(fact_5182_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B5)),A3) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),B5))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A3) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_5183_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),B5: set(A),I2: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))),B5))
     => pp(aa(set(A),bool,member(A,aa(B,A,F3,I2)),B5)) ) ).

% range_subsetD
tff(fact_5184_surj__fun__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(B,A),X6: set(B),G1: fun(A,C),G22: fun(A,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),X6) = top_top(set(A)) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),X6))
           => ( aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G1),F3),X3) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G22),F3),X3) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_5185_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).

% not_UNIV_le_Icc
tff(fact_5186_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,bool))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),S),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S))))) ) ).

% Inf_INT_eq2
tff(fact_5187_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H))) ) ).

% not_UNIV_le_Iic
tff(fact_5188_Compl__UNIV__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).

% Compl_UNIV_eq
tff(fact_5189_Compl__empty__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).

% Compl_empty_eq
tff(fact_5190_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,set(A)),S: set(B),X5: A] :
      ( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),image2(B,fun(A,bool),aTP_Lamp_yu(fun(B,set(A)),fun(B,fun(A,bool)),R3)),S)),X5))
    <=> pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R3),S)))) ) ).

% INF_INT_eq
tff(fact_5191_Compl__partition2,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A6)),A6) = top_top(set(A)) ).

% Compl_partition2
tff(fact_5192_Compl__partition,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),uminus_uminus(set(A)),A6)) = top_top(set(A)) ).

% Compl_partition
tff(fact_5193_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A6) ).

% Compl_eq_Diff_UNIV
tff(fact_5194_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: fun(C,set(product_prod(A,B))),S: set(C),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_zk(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R3)),S)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R3),S)))) ) ).

% INF_INT_eq2
tff(fact_5195_sorted__remdups,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),remdups(A),Xs)) ) ) ).

% sorted_remdups
tff(fact_5196_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P2: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_aaa(fun(C,fun(B,A)),fun(B,A),P2)),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,C)),set(A),image2(fun(B,C),A,aTP_Lamp_aac(fun(C,fun(B,A)),fun(fun(B,C),A),P2)),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_5197_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P2: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_aad(fun(C,fun(B,A)),fun(B,A),P2)),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(fun(B,C)),set(A),image2(fun(B,C),A,aTP_Lamp_aae(fun(C,fun(B,A)),fun(fun(B,C),A),P2)),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_5198_Inf__int__def,axiom,
    ! [X6: set(int)] : aa(set(int),int,complete_Inf_Inf(int),X6) = aa(int,int,uminus_uminus(int),aa(set(int),int,complete_Sup_Sup(int),aa(set(int),set(int),image2(int,int,uminus_uminus(int)),X6))) ).

% Inf_int_def
tff(fact_5199_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G3: fun(B,A),F3: fun(A,C)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,G3),top_top(set(B)))))
     => pp(aa(set(C),bool,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_aaf(fun(B,A),fun(fun(A,C),fun(B,C)),G3),F3)),top_top(set(B))))) ) ).

% finite_range_imageI
tff(fact_5200_INTER__UNIV__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A)),A6: set(B)] :
      ( ( top_top(set(A)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
         => ( aa(B,set(A),B5,X4) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_5201_INTER__UNIV__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A)),A6: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6)) = top_top(set(A)) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
         => ( aa(B,set(A),B5,X4) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_5202_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% sup_shunt
tff(fact_5203_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A3: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),X) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),X) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Y) = top_top(A) )
               => ( X = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_5204_top_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).

% top.ordering_top_axioms
tff(fact_5205_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: A,X: B] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))) )
     => ( aa(B,A,F3,X) = A3 ) ) ).

% range_eq_singletonD
tff(fact_5206_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_5207_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),C3: A] :
          ( ( ( A6 = bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wq(A,fun(B,A),C3)),A6)) = top_top(A) ) )
          & ( ( A6 != bot_bot(set(B)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_wq(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).

% INF_constant
tff(fact_5208_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),uminus_uminus(set(B)),A6)))) ) ).

% surj_Compl_image_subset
tff(fact_5209_length__remdups__card,axiom,
    ! [A: $tType,L: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),L)) = aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),L)) ).

% length_remdups_card
tff(fact_5210_finite__range__Some,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_range_Some
tff(fact_5211_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ pp(aa(set(option(A)),bool,member(option(A),X),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_5212_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),A3: B,B2: A] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F3),top_top(set(B)))))
     => pp(aa(set(option(A)),bool,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),fun_upd(B,option(A),F3,A3,aa(A,option(A),some(A),B2))),top_top(set(B))))) ) ).

% finite_range_updI
tff(fact_5213_INT__empty,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_5214_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => folding_insort_key(A,A,ord_less_eq(A),ord_less(A),top_top(set(A)),aTP_Lamp_pk(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_5215_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),X) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_5216_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_5217_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),aa(list(A),list(A),remdups(A),Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_5218_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).

% finite_UNIV_card_ge_0
tff(fact_5219_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R2: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G3),top_top(set(C)))),S))
       => finite4664212375090638736ute_on(C,B,R2,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_5220_INT__extend__simps_I3_J,axiom,
    ! [F2: $tType,E: $tType,C4: set(E),A6: fun(E,set(F2)),B5: set(F2)] :
      ( ( ( C4 = bot_bot(set(E)) )
       => ( aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),A6),C4))),B5) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),top_top(set(F2))),B5) ) )
      & ( ( C4 != bot_bot(set(E)) )
       => ( aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),A6),C4))),B5) = aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_zw(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A6),B5)),C4)) ) ) ) ).

% INT_extend_simps(3)
tff(fact_5221_UN__UN__finite__eq,axiom,
    ! [A: $tType,A6: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aTP_Lamp_aag(fun(nat,set(A)),fun(nat,set(A)),A6)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A6),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_5222_remdup__sort__mergesort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),remdups(A)),linorder_sort_key(A,A,aTP_Lamp_pk(A,A))) = mergesort_remdups(A) ) ) ).

% remdup_sort_mergesort_remdups
tff(fact_5223_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))))) ) ).

% card_range_greater_zero
tff(fact_5224_UN__finite__subset,axiom,
    ! [A: $tType,A6: fun(nat,set(A)),C4: set(A)] :
      ( ! [N4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N4)))),C4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A6),top_top(set(nat))))),C4)) ) ).

% UN_finite_subset
tff(fact_5225_UN__finite2__eq,axiom,
    ! [A: $tType,A6: fun(nat,set(A)),B5: fun(nat,set(A)),K2: nat] :
      ( ! [N4: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K2))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A6),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_5226_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_aah(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_5227_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,product_prod(A,C))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,aa(fun(B,product_prod(A,C)),fun(B,A),comp(product_prod(A,C),A,B,product_fst(A,C)),F3)),top_top(set(B)))))
     => ( pp(aa(set(C),bool,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(B,product_prod(A,C)),fun(B,C),comp(product_prod(A,C),C,B,product_snd(A,C)),F3)),top_top(set(B)))))
       => pp(aa(set(product_prod(A,C)),bool,finite_finite2(product_prod(A,C)),aa(set(B),set(product_prod(A,C)),image2(B,product_prod(A,C),F3),top_top(set(B))))) ) ) ).

% finite_range_prod
tff(fact_5228_UN__finite2__subset,axiom,
    ! [A: $tType,A6: fun(nat,set(A)),B5: fun(nat,set(A)),K2: nat] :
      ( ! [N4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N4)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K2))))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A6),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),top_top(set(nat)))))) ) ).

% UN_finite2_subset
tff(fact_5229_UN__constant__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,A6: set(A),F3: fun(A,set(B)),C3: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => ( aa(A,set(B),F3,X3) = C3 ) )
       => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),A6)) = C3 ) ) ) ).

% UN_constant_eq
tff(fact_5230_Sup__option__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(option(A))] :
          ( ( ( ( A6 = bot_bot(set(option(A))) )
              | ( A6 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) )
           => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A6) = none(A) ) )
          & ( ~ ( ( A6 = bot_bot(set(option(A))) )
                | ( A6 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) )
           => ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A6) = aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),these(A,A6))) ) ) ) ) ).

% Sup_option_def
tff(fact_5231_Pow__fold,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pow(A,A6) = finite_fold(A,set(set(A)),aTP_Lamp_wg(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A6) ) ) ).

% Pow_fold
tff(fact_5232_INF__filter__not__bot,axiom,
    ! [I6: $tType,A: $tType,B5: set(I6),F4: fun(I6,filter(A))] :
      ( ! [X10: set(I6)] :
          ( pp(aa(set(I6),bool,aa(set(I6),fun(set(I6),bool),ord_less_eq(set(I6)),X10),B5))
         => ( pp(aa(set(I6),bool,finite_finite2(I6),X10))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image2(I6,filter(A),F4),X10)) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image2(I6,filter(A),F4),B5)) != bot_bot(filter(A)) ) ) ).

% INF_filter_not_bot
tff(fact_5233_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_5234_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_5235_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_5236_Pow__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(set(A)),bool,member(set(A),A6),pow(A,B5)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ).

% Pow_iff
tff(fact_5237_PowI,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(set(A)),bool,member(set(A),A6),pow(A,B5))) ) ).

% PowI
tff(fact_5238_Pow__UNIV,axiom,
    ! [A: $tType] : pow(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_5239_mod__true,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(top_top(assn)),H))
    <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,H)) ) ).

% mod_true
tff(fact_5240_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: bool] :
      ( ( pp(P2)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_aai(bool,fun(A,fun(B,bool)),P2))) = top_top(set(product_prod(A,B))) ) )
      & ( ~ pp(P2)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_aai(bool,fun(A,fun(B,bool)),P2))) = bot_bot(set(product_prod(A,B))) ) ) ) ).

% Collect_const_case_prod
tff(fact_5241_these__image__Some__eq,axiom,
    ! [A: $tType,A6: set(A)] : these(A,aa(set(A),set(option(A)),image2(A,option(A),some(A)),A6)) = A6 ).

% these_image_Some_eq
tff(fact_5242_Pow__Int__eq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) = 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,A6)),pow(A,B5)) ).

% Pow_Int_eq
tff(fact_5243_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_5244_these__insert__None,axiom,
    ! [A: $tType,A6: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),A6)) = these(A,A6) ).

% these_insert_None
tff(fact_5245_Pow__empty,axiom,
    ! [A: $tType] : pow(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_5246_Pow__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A)] :
      ( ( pow(A,X6) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Y6),bot_bot(set(set(A)))) )
    <=> ( ( X6 = bot_bot(set(A)) )
        & ( Y6 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_5247_these__insert__Some,axiom,
    ! [A: $tType,X: A,A6: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),aa(A,option(A),some(A),X)),A6)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),these(A,A6)) ).

% these_insert_Some
tff(fact_5248_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F4: fun(A,filter(B))] :
      ( ! [I3: A] :
          ( pp(aa(set(A),bool,member(A,I3),I5))
         => ! [J3: A] :
              ( pp(aa(set(A),bool,member(A,J3),I5))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),I5))
                  & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F4,X5)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I3)),aa(A,filter(B),F4,J3)))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I5)) = bot_bot(filter(B)) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),I5))
            & ( aa(A,filter(B),F4,X4) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_5249_UNIV__unit,axiom,
    top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),aa(product_unit,fun(set(product_unit),set(product_unit)),insert(product_unit),product_Unity),bot_bot(set(product_unit))) ).

% UNIV_unit
tff(fact_5250_Pow__bottom,axiom,
    ! [A: $tType,B5: set(A)] : pp(aa(set(set(A)),bool,member(set(A),bot_bot(set(A))),pow(A,B5))) ).

% Pow_bottom
tff(fact_5251_Pow__not__empty,axiom,
    ! [A: $tType,A6: set(A)] : pow(A,A6) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_5252_Inf__filter__not__bot,axiom,
    ! [A: $tType,B5: set(filter(A))] :
      ( ! [X10: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X10),B5))
         => ( pp(aa(set(filter(A)),bool,finite_finite2(filter(A)),X10))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X10) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_5253_ent__true,axiom,
    ! [P2: assn] : entails(P2,top_top(assn)) ).

% ent_true
tff(fact_5254_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,bool),set(A),collect(A),top_top(fun(A,bool))) ).

% top_set_def
tff(fact_5255_top__unit__def,axiom,
    top_top(product_unit) = product_Unity ).

% top_unit_def
tff(fact_5256_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),top_top(set(product_prod(A,B))))) ) ).

% top_empty_eq2
tff(fact_5257_Pow__top,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(set(A)),bool,member(set(A),A6),pow(A,A6))) ).

% Pow_top
tff(fact_5258_PowD,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(set(A)),bool,member(set(A),A6),pow(A,B5)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ).

% PowD
tff(fact_5259_Pow__def,axiom,
    ! [A: $tType,A6: set(A)] : pow(A,A6) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ls(set(A),fun(set(A),bool),A6)) ).

% Pow_def
tff(fact_5260_in__these__eq,axiom,
    ! [A: $tType,X: A,A6: set(option(A))] :
      ( pp(aa(set(A),bool,member(A,X),these(A,A6)))
    <=> pp(aa(set(option(A)),bool,member(option(A),aa(A,option(A),some(A),X)),A6)) ) ).

% in_these_eq
tff(fact_5261_mod__star__trueI,axiom,
    ! [P2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H))
     => pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),top_top(assn))),H)) ) ).

% mod_star_trueI
tff(fact_5262_mod__star__trueE,axiom,
    ! [P2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),top_top(assn))),H))
     => ~ ! [H4: product_prod(heap_ext(product_unit),set(nat))] : ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H4)) ) ).

% mod_star_trueE
tff(fact_5263_Pow__mono,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pow(A,A6)),pow(A,B5))) ) ).

% Pow_mono
tff(fact_5264_subset__Pow__Union,axiom,
    ! [A: $tType,A6: set(set(A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A6),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6)))) ).

% subset_Pow_Union
tff(fact_5265_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B),B5: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A6) = B5 )
     => ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),pow(B,A6)) = pow(A,B5) ) ) ).

% image_Pow_surj
tff(fact_5266_top__assn__def,axiom,
    top_top(assn) = abs_assn(in_range) ).

% top_assn_def
tff(fact_5267_SUP__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: fun(bool,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(bool),set(A),image2(bool,A,A6),top_top(set(bool)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(bool,A,A6,fTrue)),aa(bool,A,A6,fFalse)) ) ).

% SUP_UNIV_bool_expand
tff(fact_5268_INF__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: fun(bool,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(bool),set(A),image2(bool,A,A6),top_top(set(bool)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(bool,A,A6,fTrue)),aa(bool,A,A6,fFalse)) ) ).

% INF_UNIV_bool_expand
tff(fact_5269_UN__bool__eq,axiom,
    ! [A: $tType,A6: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(bool),set(set(A)),image2(bool,set(A),A6),top_top(set(bool)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(bool,set(A),A6,fTrue)),aa(bool,set(A),A6,fFalse)) ).

% UN_bool_eq
tff(fact_5270_Un__eq__UN,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(bool),set(set(A)),image2(bool,set(A),aa(set(A),fun(bool,set(A)),aTP_Lamp_aaj(set(A),fun(set(A),fun(bool,set(A))),A6),B5)),top_top(set(bool)))) ).

% Un_eq_UN
tff(fact_5271_INT__bool__eq,axiom,
    ! [A: $tType,A6: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(bool),set(set(A)),image2(bool,set(A),A6),top_top(set(bool)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(bool,set(A),A6,fTrue)),aa(bool,set(A),A6,fFalse)) ).

% INT_bool_eq
tff(fact_5272_Pow__INT__eq,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A6: set(B)] : pow(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))) = aa(set(set(set(A))),set(set(A)),complete_Inf_Inf(set(set(A))),aa(set(B),set(set(set(A))),image2(B,set(set(A)),aTP_Lamp_aak(fun(B,set(A)),fun(B,set(set(A))),B5)),A6)) ).

% Pow_INT_eq
tff(fact_5273_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_ext(product_unit)] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(top_top(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))) ).

% mod_h_bot_iff(2)
tff(fact_5274_Un__Pow__subset,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),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,A6)),pow(A,B5))),pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))) ).

% Un_Pow_subset
tff(fact_5275_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A6: set(B)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),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))),image2(B,set(set(A)),aTP_Lamp_aak(fun(B,set(A)),fun(B,set(set(A))),B5)),A6))),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6))))) ).

% UN_Pow_subset
tff(fact_5276_Pow__insert,axiom,
    ! [A: $tType,A3: A,A6: set(A)] : pow(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6)) = 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,A6)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3)),pow(A,A6))) ).

% Pow_insert
tff(fact_5277_Option_Othese__def,axiom,
    ! [A: $tType,A6: set(option(A))] : these(A,A6) = aa(set(option(A)),set(A),image2(option(A),A,the2(A)),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_aal(set(option(A)),fun(option(A),bool),A6))) ).

% Option.these_def
tff(fact_5278_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A6)),B5))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),pow(B,A6))),pow(A,B5))) ) ).

% image_Pow_mono
tff(fact_5279_binomial__def,axiom,
    ! [N: nat,K2: nat] : aa(nat,nat,binomial(N),K2) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),bool),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),bool),aTP_Lamp_aam(nat,fun(nat,fun(set(nat),bool)),N),K2))) ).

% binomial_def
tff(fact_5280_mod__star__trueE_H,axiom,
    ! [P2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),top_top(assn))),H))
     => ~ ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
            ( ( aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H4) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H) )
           => ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H4)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H)))
             => ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),H4)) ) ) ) ).

% mod_star_trueE'
tff(fact_5281_Some__image__these__eq,axiom,
    ! [A: $tType,A6: set(option(A))] : aa(set(A),set(option(A)),image2(A,option(A),some(A)),these(A,A6)) = aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_aal(set(option(A)),fun(option(A),bool),A6)) ).

% Some_image_these_eq
tff(fact_5282_Pow__set_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : pow(B,aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs))) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),pow(B,aa(list(B),set(B),set2(B),Xs))),aa(set(set(B)),set(set(B)),image2(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X)),pow(B,aa(list(B),set(B),set2(B),Xs)))) ).

% Pow_set(2)
tff(fact_5283_Inf__option__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(option(A))] :
          ( ( pp(aa(set(option(A)),bool,member(option(A),none(A)),A6))
           => ( aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),A6) = none(A) ) )
          & ( ~ pp(aa(set(option(A)),bool,member(option(A),none(A)),A6))
           => ( aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),A6) = aa(A,option(A),some(A),aa(set(A),A,complete_Inf_Inf(A),these(A,A6))) ) ) ) ) ).

% Inf_option_def
tff(fact_5284_these__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) = bot_bot(set(A)) )
    <=> ( ( B5 = bot_bot(set(option(A))) )
        | ( B5 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_5285_these__not__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) != bot_bot(set(A)) )
    <=> ( ( B5 != bot_bot(set(option(A))) )
        & ( B5 != aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_5286_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A6)),B5))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),finite_Fpow(B,A6))),finite_Fpow(A,B5))) ) ).

% image_Fpow_mono
tff(fact_5287_Un__set__drop__extend,axiom,
    ! [A: $tType,J2: nat,L: list(set(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),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,aa(nat,fun(nat,nat),minus_minus(nat),J2),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),J2,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,aa(nat,fun(nat,nat),minus_minus(nat),J2),aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).

% Un_set_drop_extend
tff(fact_5288_relInvImage__UNIV__relImage,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F3: fun(A,B)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),bNF_Gr7122648621184425601vImage(A,B,top_top(set(A)),bNF_Gr4221423524335903396lImage(A,B,R2,F3),F3))) ).

% relInvImage_UNIV_relImage
tff(fact_5289_top1I,axiom,
    ! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).

% top1I
tff(fact_5290_drop__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : drop(A,N,drop(A,M,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs) ).

% drop_drop
tff(fact_5291_top2I,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X),Y)) ).

% top2I
tff(fact_5292_drop__upd__irrelevant,axiom,
    ! [A: $tType,M: nat,N: nat,L: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( drop(A,N,list_update(A,L,M,X)) = drop(A,N,L) ) ) ).

% drop_upd_irrelevant
tff(fact_5293_drop__update__cancel,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( drop(A,M,list_update(A,Xs,N,X)) = drop(A,M,Xs) ) ) ).

% drop_update_cancel
tff(fact_5294_drop__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,N,Xs) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_5295_drop__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( drop(A,N,Xs) = nil(A) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_5296_drop__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( drop(A,N,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_5297_nth__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,drop(A,N,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I2)) ) ) ).

% nth_drop
tff(fact_5298_UNIV__bool,axiom,
    top_top(set(bool)) = aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fFalse),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))) ).

% UNIV_bool
tff(fact_5299_less__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less(filter(A)),F4),F6))
    <=> ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F6))
        & ~ pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F6),F4)) ) ) ).

% less_filter_def
tff(fact_5300_drop__eq__ConsD,axiom,
    ! [A: $tType,N: nat,Xs: list(A),X: A,Xs5: list(A)] :
      ( ( drop(A,N,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs5) )
     => ( drop(A,aa(nat,nat,suc,N),Xs) = Xs5 ) ) ).

% drop_eq_ConsD
tff(fact_5301_set__drop__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,N,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_drop_subset
tff(fact_5302_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,N,Xs)) ) ) ).

% sorted_drop
tff(fact_5303_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_5304_relImage__mono,axiom,
    ! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
     => pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr4221423524335903396lImage(A,B,R1,F3)),bNF_Gr4221423524335903396lImage(A,B,R22,F3))) ) ).

% relImage_mono
tff(fact_5305_set__drop__subset__set__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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,N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_5306_drop__update__swap,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( drop(A,M,list_update(A,Xs,N,X)) = list_update(A,drop(A,M,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M),X) ) ) ).

% drop_update_swap
tff(fact_5307_drop__takeWhile,axiom,
    ! [A: $tType,I2: nat,P2: fun(A,bool),L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,L))))
     => ( drop(A,I2,takeWhile(A,P2,L)) = takeWhile(A,P2,drop(A,I2,L)) ) ) ).

% drop_takeWhile
tff(fact_5308_drop__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aTP_Lamp_aan(list(A),fun(nat,list(A)),Xs),N) ).

% drop_Cons
tff(fact_5309_nths__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I5: set(nat)] : nths(A,drop(A,N,Xs),I5) = nths(A,Xs,aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),I5)) ).

% nths_drop
tff(fact_5310_hd__drop__conv__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),A,hd(A),drop(A,N,Xs)) = aa(nat,A,nth(A,Xs),N) ) ) ).

% hd_drop_conv_nth
tff(fact_5311_Fpow__mono,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A6)),finite_Fpow(A,B5))) ) ).

% Fpow_mono
tff(fact_5312_Fpow__subset__Pow,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A6)),pow(A,A6))) ).

% Fpow_subset_Pow
tff(fact_5313_Fpow__def,axiom,
    ! [A: $tType,A6: set(A)] : finite_Fpow(A,A6) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_aao(set(A),fun(set(A),bool),A6)) ).

% Fpow_def
tff(fact_5314_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs)) = drop(A,I2,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_5315_in__set__drop__conv__nth,axiom,
    ! [A: $tType,X: A,N: nat,L: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),drop(A,N,L))))
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),I))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
          & ( X = aa(nat,A,nth(A,L),I) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_5316_Fpow__Pow__finite,axiom,
    ! [A: $tType,A6: set(A)] : finite_Fpow(A,A6) = 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,A6)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_5317_drop__last__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ( drop(A,aa(nat,nat,aa(nat,fun(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),aa(A,fun(list(A),list(A)),cons(A),last(A,L)),nil(A)) ) ) ).

% drop_last_conv
tff(fact_5318_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,N,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aaq(nat,fun(nat,bool),N)))) ).

% ntrancl_def
tff(fact_5319_take__hd__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),drop(A,N,Xs))),nil(A))) = take(A,aa(nat,nat,suc,N),Xs) ) ) ).

% take_hd_drop
tff(fact_5320_drop__upt,axiom,
    ! [M: nat,I2: nat,J2: nat] : drop(nat,M,upt(I2,J2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M),J2) ).

% drop_upt
tff(fact_5321_take__update,axiom,
    ! [A: $tType,N: nat,L: list(A),I2: nat,X: A] : take(A,N,list_update(A,L,I2,X)) = list_update(A,take(A,N,L),I2,X) ).

% take_update
tff(fact_5322_take__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( take(A,N,Xs) = Xs ) ) ).

% take_all
tff(fact_5323_take__all__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = Xs )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% take_all_iff
tff(fact_5324_take__upt,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)),N))
     => ( take(nat,M,upt(I2,N)) = upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)) ) ) ).

% take_upt
tff(fact_5325_nth__take,axiom,
    ! [A: $tType,I2: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
     => ( aa(nat,A,nth(A,take(A,N,Xs)),I2) = aa(nat,A,nth(A,Xs),I2) ) ) ).

% nth_take
tff(fact_5326_Misc_Olast__in__set,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => pp(aa(set(A),bool,member(A,last(A,L)),aa(list(A),set(A),set2(A),L))) ) ).

% Misc.last_in_set
tff(fact_5327_take__update__cancel,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( take(A,N,list_update(A,Xs,M,Y)) = take(A,N,Xs) ) ) ).

% take_update_cancel
tff(fact_5328_last__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( last(nat,upt(I2,J2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),one_one(nat)) ) ) ).

% last_upt
tff(fact_5329_hd__take,axiom,
    ! [A: $tType,J2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J2))
     => ( aa(list(A),A,hd(A),take(A,J2,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_take
tff(fact_5330_last__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( last(A,drop(A,N,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_5331_finite__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% finite_relpow
tff(fact_5332_set__take__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,N,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_take_subset
tff(fact_5333_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,N,Xs)) ) ) ).

% sorted_take
tff(fact_5334_relpow__Suc__D2_H,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A)),X5: A,Y5: A,Z6: A] :
      ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z6)),R2)) )
     => ? [W: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),W)),R2))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z6)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_D2'
tff(fact_5335_take__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : take(A,N,drop(A,M,Xs)) = drop(A,M,take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs)) ).

% take_drop
tff(fact_5336_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys2)) ) ) ) ) ).

% last_zip
tff(fact_5337_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2)))
     => ( X = Y ) ) ).

% relpow_0_E
tff(fact_5338_relpow__0__I,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2))) ).

% relpow_0_I
tff(fact_5339_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z4: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),R2)) ) ) ).

% relpow_Suc_E
tff(fact_5340_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y: A,N: nat,R2: set(product_prod(A,A)),Z4: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),R2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2))) ) ) ).

% relpow_Suc_I
tff(fact_5341_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z4: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
     => ? [Y3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_D2
tff(fact_5342_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z4: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_E2
tff(fact_5343_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z4: A,N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2))) ) ) ).

% relpow_Suc_I2
tff(fact_5344_last__filter,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool)] :
      ( ( Xs != nil(A) )
     => ( pp(aa(A,bool,P2,last(A,Xs)))
       => ( last(A,filter2(A,P2,Xs)) = last(A,Xs) ) ) ) ).

% last_filter
tff(fact_5345_tl__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) != nil(A) )
     => ( last(A,Xs) = last(A,aa(list(A),list(A),tl(A),Xs)) ) ) ).

% tl_last
tff(fact_5346_relpow__add,axiom,
    ! [A: $tType,M: nat,N: nat,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R2) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)) ).

% relpow_add
tff(fact_5347_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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,N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_5348_relpowp__relpow__eq,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A)),X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)),X5),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ).

% relpowp_relpow_eq
tff(fact_5349_last__take__nth__conv,axiom,
    ! [A: $tType,N: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L)))
     => ( ( N != zero_zero(nat) )
       => ( last(A,take(A,N,L)) = aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% last_take_nth_conv
tff(fact_5350_drop__take__drop__unsplit,axiom,
    ! [A: $tType,I2: nat,J2: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,I2,take(A,J2,L))),drop(A,J2,L)) = drop(A,I2,L) ) ) ).

% drop_take_drop_unsplit
tff(fact_5351_take__add,axiom,
    ! [A: $tType,I2: nat,J2: nat,Xs: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),take(A,J2,drop(A,I2,Xs))) ).

% take_add
tff(fact_5352_slice__def,axiom,
    ! [A: $tType,From: nat,To: nat,List: list(A)] : slice(A,From,To,List) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From),drop(A,From,List)) ).

% slice_def
tff(fact_5353_relpow__E,axiom,
    ! [A: $tType,X: A,Z4: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z4 ) )
       => ~ ! [Y3: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),R2)) ) ) ) ) ).

% relpow_E
tff(fact_5354_relpow__E2,axiom,
    ! [A: $tType,X: A,Z4: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z4 ) )
       => ~ ! [Y3: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R2))) ) ) ) ) ).

% relpow_E2
tff(fact_5355_trancl__power,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_trancl(A,R2)))
    <=> ? [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N3),R2))) ) ) ).

% trancl_power
tff(fact_5356_relpow__empty,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_5357_nth__take__lemma,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Ys2: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Ys2)))
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K2))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys2),I3) ) )
         => ( take(A,K2,Xs) = take(A,K2,Ys2) ) ) ) ) ).

% nth_take_lemma
tff(fact_5358_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),Xs_2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys_1),Ys_2) )
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1)),Ys_2) ) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),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_5359_filter__upt__take__conv,axiom,
    ! [A: $tType,P2: fun(A,bool),M: nat,L: list(A),N: nat] : filter2(nat,aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_aar(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),P2),M),L),upt(N,M)) = filter2(nat,aa(list(A),fun(nat,bool),aTP_Lamp_tb(fun(A,bool),fun(list(A),fun(nat,bool)),P2),L),upt(N,M)) ).

% filter_upt_take_conv
tff(fact_5360_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P2: fun(A,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I3))) ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
         => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),N))) )
       => ( takeWhile(A,P2,Xs) = take(A,N,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_5361_take__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_aas(A,fun(list(A),fun(nat,list(A))),X),Xs),N) ).

% take_Cons
tff(fact_5362_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N: nat,X: A,Ys2: list(B)] : zip(A,B,replicate(A,N,X),Ys2) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),take(B,N,Ys2)) ).

% zip_replicate1
tff(fact_5363_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
    <=> ? [F11: fun(nat,A)] :
          ( ( aa(nat,A,F11,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F11,N) = B2 )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
             => pp(aa(set(product_prod(A,A)),bool,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,F11,I)),aa(nat,A,F11,aa(nat,nat,suc,I)))),R2)) ) ) ) ).

% relpow_fun_conv
tff(fact_5364_sorted__hd__last,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( ( L != nil(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,hd(A),L)),last(A,L))) ) ) ) ).

% sorted_hd_last
tff(fact_5365_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : zip(A,B,Xs,replicate(B,N,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_oz(B,fun(A,product_prod(A,B))),Y)),take(A,N,Xs)) ).

% zip_replicate2
tff(fact_5366_hd__last__singletonI,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( aa(list(A),A,hd(A),Xs) = last(A,Xs) )
       => ( distinct(A,Xs)
         => ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),nil(A)) ) ) ) ) ).

% hd_last_singletonI
tff(fact_5367_filter__nth__ex__nth,axiom,
    ! [A: $tType,N: nat,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))))
     => ? [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M5))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,filter2(A,P2,Xs)),N) = aa(nat,A,nth(A,Xs),M5) )
          & ( filter2(A,P2,take(A,M5,Xs)) = take(A,N,filter2(A,P2,Xs)) ) ) ) ).

% filter_nth_ex_nth
tff(fact_5368_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,X: A,Ys2: list(B),Xs: list(A),F3: fun(A,option(B)),Y: B] :
      ( ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys2) = map_upds(A,B,F3,Xs,Ys2) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys2) = fun_upd(A,option(B),map_upds(A,B,F3,Xs,Ys2),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_5369_Union__take__drop__id,axiom,
    ! [A: $tType,N: nat,L: list(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(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),N,L)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),take(set(A),N,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),L)) ).

% Union_take_drop_id
tff(fact_5370_relpow__finite__bounded,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K2: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K2),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aat(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ).

% relpow_finite_bounded
tff(fact_5371_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I2: nat,J2: nat] :
      ( distinct(A,Vs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => ( 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,I2,Vs))),aa(list(A),set(A),set2(A),drop(A,J2,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_5372_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),lex(A,R3)))
     => ~ ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys2)))
             => ( ( take(A,I3,Xs) = take(A,I3,Ys2) )
               => ~ pp(aa(set(product_prod(A,A)),bool,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),I3)),aa(nat,A,nth(A,Ys2),I3))),R3)) ) ) ) ) ).

% lex_take_index
tff(fact_5373_foldl__list__update,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),F3: fun(B,fun(A,B)),A3: B,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),A3),list_update(A,Xs,N,X)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),aa(A,B,aa(B,fun(A,B),F3,aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),A3),take(A,N,Xs))),X)),drop(A,aa(nat,nat,suc,N),Xs)) ) ) ).

% foldl_list_update
tff(fact_5374_map__nth__upt__drop__take__conv,axiom,
    ! [A: $tType,N7: nat,L: list(A),M6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),aa(list(A),nat,size_size(list(A)),L)))
     => ( aa(list(nat),list(A),map(nat,A,nth(A,L)),upt(M6,N7)) = drop(A,M6,take(A,N7,L)) ) ) ).

% map_nth_upt_drop_take_conv
tff(fact_5375_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,aa(nat,nat,suc,I2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_5376_take__update__last,axiom,
    ! [A: $tType,N: nat,List: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),List)))
     => ( list_update(A,take(A,aa(nat,nat,suc,N),List),N,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,List)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ) ).

% take_update_last
tff(fact_5377_id__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_5378_mset__zip__take__Cons__drop__twice,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),J2: nat,X: A,Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( mset(product_prod(A,B),zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,J2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),drop(A,J2,Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,J2,Ys2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),drop(B,J2,Ys2))))) = aa(multiset(product_prod(A,B)),multiset(product_prod(A,B)),aa(product_prod(A,B),fun(multiset(product_prod(A,B)),multiset(product_prod(A,B))),add_mset(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),mset(product_prod(A,B),zip(A,B,Xs,Ys2))) ) ) ) ).

% mset_zip_take_Cons_drop_twice
tff(fact_5379_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),A3: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,Xs,I2,A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_5380_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(set(nat),set(A),image2(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_5381_relpow__finite__bounded1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K2: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K2),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aau(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_5382_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( transitive_trancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aau(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_5383_butlast__upd__last__eq,axiom,
    ! [A: $tType,L: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(list(A),nat,size_size(list(A)),L)))
     => ( list_update(A,butlast(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),L)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ) ).

% butlast_upd_last_eq
tff(fact_5384_range__prod,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),F3),top_top(set(C)))),product_Sigma(A,B,aa(set(C),set(A),image2(C,A,aa(fun(C,product_prod(A,B)),fun(C,A),comp(product_prod(A,B),A,C,product_fst(A,B)),F3)),top_top(set(C))),aTP_Lamp_aav(fun(C,product_prod(A,B)),fun(A,set(B)),F3)))) ).

% range_prod
tff(fact_5385_fun_Oin__rel,axiom,
    ! [A: $tType,B: $tType,D: $tType,R2: fun(A,fun(B,bool)),A3: fun(D,A),B2: fun(D,B)] :
      ( pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),R2),A3),B2))
    <=> ? [Z2: fun(D,product_prod(A,B))] :
          ( pp(aa(set(fun(D,product_prod(A,B))),bool,member(fun(D,product_prod(A,B)),Z2),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_aaw(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R2))))
          & ( aa(fun(D,product_prod(A,B)),fun(D,A),comp(product_prod(A,B),A,D,product_fst(A,B)),Z2) = A3 )
          & ( aa(fun(D,product_prod(A,B)),fun(D,B),comp(product_prod(A,B),B,D,product_snd(A,B)),Z2) = B2 ) ) ) ).

% fun.in_rel
tff(fact_5386_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B5)))
    <=> ( pp(aa(set(A),bool,member(A,A3),A6))
        & pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B5,A3))) ) ) ).

% mem_Sigma_iff
tff(fact_5387_SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B5,A3)))
       => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B5))) ) ) ).

% SigmaI
tff(fact_5388_Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),Q2: fun(B,bool)] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_aax(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),P2),Q2))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P2),aTP_Lamp_aay(fun(B,bool),fun(A,set(B)),Q2)) ).

% Collect_case_prod
tff(fact_5389_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B5: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B5) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_5390_Times__empty,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ( product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A6 = bot_bot(set(A)) )
        | ( B5 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_5391_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : product_Sigma(A,B,A6,aTP_Lamp_aba(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_5392_Sigma__UNIV__cancel,axiom,
    ! [B: $tType,A: $tType,A6: set(A),X6: set(B)] : 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))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),X6))),product_Sigma(A,B,A6,aTP_Lamp_abb(A,set(B)))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_UNIV_cancel
tff(fact_5393_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_abb(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A6),aTP_Lamp_abb(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_5394_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A6: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aaz(set(B),fun(A,set(B)),A6))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_abc(set(B),fun(A,set(B)),A6)) ).

% Compl_Times_UNIV1
tff(fact_5395_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ! [A5: A] :
            ( pp(aa(set(A),bool,member(A,A5),A6))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B5,A5))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,B5))) ) ) ).

% finite_SigmaI
tff(fact_5396_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_abb(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_5397_fst__image__times,axiom,
    ! [B: $tType,A: $tType,B5: set(B),A6: set(A)] :
      ( ( ( B5 = bot_bot(set(B)) )
       => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))) = bot_bot(set(A)) ) )
      & ( ( B5 != bot_bot(set(B)) )
       => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))) = A6 ) ) ) ).

% fst_image_times
tff(fact_5398_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A6: set(B),B5: set(A)] :
      ( ( ( A6 = bot_bot(set(B)) )
       => ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A6,aTP_Lamp_ws(set(A),fun(B,set(A)),B5))) = bot_bot(set(A)) ) )
      & ( ( A6 != bot_bot(set(B)) )
       => ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A6,aTP_Lamp_ws(set(A),fun(B,set(A)),B5))) = B5 ) ) ) ).

% snd_image_times
tff(fact_5399_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys2)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_abd(list(B),fun(A,set(B)),Ys2)) ).

% set_product
tff(fact_5400_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A3: A,A6: set(A),B2: B,B5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6),aa(set(B),fun(A,set(B)),aTP_Lamp_abe(B,fun(set(B),fun(A,set(B))),B2),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(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),A3),B2)),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))),product_Sigma(A,B,A6,aa(set(B),fun(A,set(B)),aTP_Lamp_abe(B,fun(set(B),fun(A,set(B))),B2),B5))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6),aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))) ).

% insert_Times_insert
tff(fact_5401_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B5,X3))) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A6,B5)) = groups7311177749621191930dd_sum(A,nat,aTP_Lamp_yb(fun(A,set(B)),fun(A,nat),B5),A6) ) ) ) ).

% card_SigmaI
tff(fact_5402_take__butlast__conv,axiom,
    ! [A: $tType,L: list(A)] : take(A,aa(nat,nat,aa(nat,fun(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_5403_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B),C4: set(A),D5: set(B)] :
      ( ( product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)) = product_Sigma(A,B,C4,aTP_Lamp_aaz(set(B),fun(A,set(B)),D5)) )
    <=> ( ( ( A6 = C4 )
          & ( B5 = D5 ) )
        | ( ( ( A6 = bot_bot(set(A)) )
            | ( B5 = bot_bot(set(B)) ) )
          & ( ( C4 = bot_bot(set(A)) )
            | ( D5 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_5404_less__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less(nat)),ord_less(nat))) ).

% less_natural.rsp
tff(fact_5405_less__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less(int)),ord_less(int))) ).

% less_integer.rsp
tff(fact_5406_fun_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: fun(A,fun(C,bool)),X: fun(D,A),G3: fun(B,C),Y: fun(D,B)] :
      ( pp(aa(fun(D,C),bool,aa(fun(D,A),fun(fun(D,C),bool),bNF_rel_fun(D,D,A,C,fequal(D),Sa),X),aa(fun(D,B),fun(D,C),comp(B,C,D,G3),Y)))
    <=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_abf(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G3)),X),Y)) ) ).

% fun.rel_map(2)
tff(fact_5407_fun_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: fun(D,A),Y: fun(D,B)] :
      ( pp(aa(fun(D,B),bool,aa(fun(D,C),fun(fun(D,B),bool),bNF_rel_fun(D,D,C,B,fequal(D),Sb),aa(fun(D,A),fun(D,C),comp(A,C,D,I2),X)),Y))
    <=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_abg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% fun.rel_map(1)
tff(fact_5408_Collect__case__prod__Sigma,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),Q2: fun(A,fun(B,bool))] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_abh(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P2),Q2))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P2),aTP_Lamp_abi(fun(A,fun(B,bool)),fun(A,set(B)),Q2)) ).

% Collect_case_prod_Sigma
tff(fact_5409_plus__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),plus_plus(int)),plus_plus(int))) ).

% plus_integer.rsp
tff(fact_5410_plus__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),plus_plus(nat)),plus_plus(nat))) ).

% plus_natural.rsp
tff(fact_5411_butlast__update_H,axiom,
    ! [A: $tType,L: list(A),I2: nat,X: A] : list_update(A,butlast(A,L),I2,X) = butlast(A,list_update(A,L,I2,X)) ).

% butlast_update'
tff(fact_5412_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C4: set(A),A6: set(B),B5: set(B)] :
      ( pp(aa(set(A),bool,member(A,X),C4))
     => ( ( product_Sigma(B,A,A6,aTP_Lamp_ws(set(A),fun(B,set(A)),C4)) = product_Sigma(B,A,B5,aTP_Lamp_ws(set(A),fun(B,set(A)),C4)) )
      <=> ( A6 = B5 ) ) ) ).

% Times_eq_cancel2
tff(fact_5413_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(A),C4: fun(A,set(B)),D5: fun(A,set(B))] :
      ( ( A6 = B5 )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),B5))
           => ( aa(A,set(B),C4,X3) = aa(A,set(B),D5,X3) ) )
       => ( product_Sigma(A,B,A6,C4) = product_Sigma(A,B,B5,D5) ) ) ) ).

% Sigma_cong
tff(fact_5414_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C4: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),J4),C4) = 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))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,C4)),product_Sigma(A,B,J4,C4)) ).

% Sigma_Diff_distrib1
tff(fact_5415_sub_Orsp,axiom,
    pp(aa(fun(num,fun(num,int)),bool,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),bool),bNF_rel_fun(num,num,fun(num,int),fun(num,int),fequal(num),bNF_rel_fun(num,num,int,int,fequal(num),fequal(int))),aTP_Lamp_abj(num,fun(num,int))),aTP_Lamp_abj(num,fun(num,int)))) ).

% sub.rsp
tff(fact_5416_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(A),C4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5),aTP_Lamp_aaz(set(B),fun(A,set(B)),C4)) = 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))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))) ).

% Times_Diff_distrib1
tff(fact_5417_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_abk(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B5)) = 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))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,A6)),product_Sigma(A,B,I5,B5)) ).

% Sigma_Diff_distrib2
tff(fact_5418_dup_Orsp,axiom,
    pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_abl(int,int)),aTP_Lamp_abl(int,int))) ).

% dup.rsp
tff(fact_5419_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(num,B),bool,aa(fun(num,A),fun(fun(num,B),bool),bNF_rel_fun(num,num,A,B,fequal(num),R2),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_5420_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B)))
             => ( pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,R2,R2),uminus_uminus(A)),uminus_uminus(B)))
               => pp(aa(fun(int,B),bool,aa(fun(int,A),fun(fun(int,B),bool),bNF_rel_fun(int,int,A,B,fequal(int),R2),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_5421_SigmaE,axiom,
    ! [A: $tType,B: $tType,C3: product_prod(A,B),A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),C3),product_Sigma(A,B,A6,B5)))
     => ~ ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => ! [Y3: B] :
                ( pp(aa(set(B),bool,member(B,Y3),aa(A,set(B),B5,X3)))
               => ( C3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) ) ) ) ) ).

% SigmaE
tff(fact_5422_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B5)))
     => pp(aa(set(A),bool,member(A,A3),A6)) ) ).

% SigmaD1
tff(fact_5423_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B5)))
     => pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B5,A3))) ) ).

% SigmaD2
tff(fact_5424_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B5)))
     => ~ ( pp(aa(set(A),bool,member(A,A3),A6))
         => ~ pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B5,A3))) ) ) ).

% SigmaE2
tff(fact_5425_predicate2__transferD,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),P2: fun(A,fun(C,bool)),Q2: fun(B,fun(D,bool)),A3: product_prod(A,B),A6: set(product_prod(A,B)),B2: product_prod(C,D),B5: set(product_prod(C,D))] :
      ( pp(aa(fun(B,fun(D,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),bool),bNF_rel_fun(A,B,fun(C,bool),fun(D,bool),R1,bNF_rel_fun(C,D,bool,bool,R22,fequal(bool))),P2),Q2))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),A3),A6))
       => ( pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),B2),B5))
         => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R1))))
           => ( pp(aa(set(product_prod(C,D)),bool,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),bool),ord_less_eq(set(product_prod(C,D))),B5),aa(fun(product_prod(C,D),bool),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),R22))))
             => ( pp(aa(C,bool,aa(A,fun(C,bool),P2,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(C,D),C,product_fst(C,D),B2)))
              <=> pp(aa(D,bool,aa(B,fun(D,bool),Q2,aa(product_prod(A,B),B,product_snd(A,B),A3)),aa(product_prod(C,D),D,product_snd(C,D),B2))) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_5426_fun_Orel__mono,axiom,
    ! [D: $tType,B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra2))
     => pp(aa(fun(fun(D,A),fun(fun(D,B),bool)),bool,aa(fun(fun(D,A),fun(fun(D,B),bool)),fun(fun(fun(D,A),fun(fun(D,B),bool)),bool),ord_less_eq(fun(fun(D,A),fun(fun(D,B),bool))),bNF_rel_fun(D,D,A,B,fequal(D),R2)),bNF_rel_fun(D,D,A,B,fequal(D),Ra2))) ) ).

% fun.rel_mono
tff(fact_5427_less__eq__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less_eq(nat)),ord_less_eq(nat))) ).

% less_eq_natural.rsp
tff(fact_5428_less__eq__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less_eq(int)),ord_less_eq(int))) ).

% less_eq_integer.rsp
tff(fact_5429_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A6: set(A),C4: set(A),B5: fun(A,set(B)),D5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B5,X3)),aa(A,set(B),D5,X3))) )
       => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A6,B5)),product_Sigma(A,B,C4,D5))) ) ) ).

% Sigma_mono
tff(fact_5430_Sigma__Int__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C4: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I5),J4),C4) = 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))),product_Sigma(A,B,I5,C4)),product_Sigma(A,B,J4,C4)) ).

% Sigma_Int_distrib1
tff(fact_5431_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C4: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I5),J4),C4) = 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))),product_Sigma(A,B,I5,C4)),product_Sigma(A,B,J4,C4)) ).

% Sigma_Un_distrib1
tff(fact_5432_member__product,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_product(A,B,A6,B5)))
    <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))) ) ).

% member_product
tff(fact_5433_Product__Type_Oproduct__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] : product_product(A,B,A6,B5) = product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)) ).

% Product_Type.product_def
tff(fact_5434_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C4: set(A),A6: set(B),B5: set(B)] :
      ( pp(aa(set(A),bool,member(A,X),C4))
     => ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A6,aTP_Lamp_ws(set(A),fun(B,set(A)),C4))),product_Sigma(B,A,B5,aTP_Lamp_ws(set(A),fun(B,set(A)),C4))))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5)) ) ) ).

% Times_subset_cancel2
tff(fact_5435_mem__Times__iff,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
    <=> ( pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),X)),A6))
        & pp(aa(set(B),bool,member(B,aa(product_prod(A,B),B,product_snd(A,B),X)),B5)) ) ) ).

% mem_Times_iff
tff(fact_5436_in__prod__fst__sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A6: set(A),B5: set(B)] :
      ( pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),X)),A6))
     => ( pp(aa(set(B),bool,member(B,aa(product_prod(A,B),B,product_snd(A,B),X)),B5))
       => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))) ) ) ).

% in_prod_fst_sndI
tff(fact_5437_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B5)) ).

% card_cartesian_product
tff(fact_5438_swap__product,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B5: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(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_oz(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A6,aTP_Lamp_ws(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),A6)) ).

% swap_product
tff(fact_5439_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I5: set(A),X6: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I5,X6) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),I5))
         => ( aa(A,set(B),X6,X4) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_5440_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B),C4: set(A),D5: set(B)] : 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))),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))),product_Sigma(A,B,C4,aTP_Lamp_aaz(set(B),fun(A,set(B)),D5))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C4),aa(set(B),fun(A,set(B)),aTP_Lamp_abm(set(B),fun(set(B),fun(A,set(B))),B5),D5)) ).

% Times_Int_Times
tff(fact_5441_Sigma__Int__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_abn(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B5)) = 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))),product_Sigma(A,B,I5,A6)),product_Sigma(A,B,I5,B5)) ).

% Sigma_Int_distrib2
tff(fact_5442_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(A),C4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5),aTP_Lamp_aaz(set(B),fun(A,set(B)),C4)) = 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))),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))) ).

% Times_Int_distrib1
tff(fact_5443_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),B5))
       => ~ pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))) ) ) ).

% infinite_cartesian_product
tff(fact_5444_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))) ) ) ).

% finite_cartesian_product
tff(fact_5445_trancl__subset__Sigma,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% trancl_subset_Sigma
tff(fact_5446_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_abp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B5)) = 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))),product_Sigma(A,B,I5,A6)),product_Sigma(A,B,I5,B5)) ).

% Sigma_Un_distrib2
tff(fact_5447_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(A),C4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5),aTP_Lamp_aaz(set(B),fun(A,set(B)),C4)) = 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))),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))) ).

% Times_Un_distrib1
tff(fact_5448_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,B)),A6: set(A),B5: set(B),S2: set(product_prod(B,C)),C4: set(C)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),bool),ord_less_eq(set(product_prod(B,C))),S2),product_Sigma(B,C,B5,aTP_Lamp_abq(set(C),fun(B,set(C)),C4))))
       => pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R3,S2)),product_Sigma(A,C,A6,aTP_Lamp_abr(set(C),fun(A,set(C)),C4)))) ) ) ).

% relcomp_subset_Sigma
tff(fact_5449_Sigma__Union,axiom,
    ! [B: $tType,A: $tType,X6: set(set(A)),B5: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X6),B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),image2(set(A),set(product_prod(A,B)),aTP_Lamp_abs(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B5)),X6)) ).

% Sigma_Union
tff(fact_5450_rev__butlast__is__tl__rev,axiom,
    ! [A: $tType,L: list(A)] : rev(A,butlast(A,L)) = aa(list(A),list(A),tl(A),rev(A,L)) ).

% rev_butlast_is_tl_rev
tff(fact_5451_Id__on__subset__Times,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id_on(A,A6)),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ).

% Id_on_subset_Times
tff(fact_5452_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,A6: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aTP_Lamp_aaz(set(B),fun(A,set(B)),A6))) = aa(set(B),nat,finite_card(B),A6) ).

% card_cartesian_product_singleton
tff(fact_5453_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A6: set(A),C4: set(B),B5: set(A),D5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),D5))))
    <=> ( ( A6 = bot_bot(set(A)) )
        | ( C4 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),D5)) ) ) ) ).

% times_subset_iff
tff(fact_5454_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A6: set(C),B5: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_rv(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),product_Sigma(C,D,A6,aTP_Lamp_abt(set(D),fun(C,set(D)),B5))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F3),A6),aa(set(D),fun(A,set(B)),aTP_Lamp_abu(fun(D,B),fun(set(D),fun(A,set(B))),G3),B5)) ).

% image_paired_Times
tff(fact_5455_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_abv(set(A),fun(fun(A,set(B)),fun(A,bool)),A6),B5))))
     => ( ! [A5: A] :
            ( pp(aa(set(A),bool,member(A,A5),A6))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B5,A5))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,B5))) ) ) ).

% finite_SigmaI2
tff(fact_5456_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
     => ( ( B5 != bot_bot(set(B)) )
       => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% finite_cartesian_productD1
tff(fact_5457_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
     => ( ( A6 != bot_bot(set(A)) )
       => pp(aa(set(B),bool,finite_finite2(B),B5)) ) ) ).

% finite_cartesian_productD2
tff(fact_5458_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
    <=> ( ( A6 = bot_bot(set(A)) )
        | ( B5 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,finite_finite2(A),A6))
          & pp(aa(set(B),bool,finite_finite2(B),B5)) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_5459_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A6,B5)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_abv(set(A),fun(fun(A,set(B)),fun(A,bool)),A6),B5)) ).

% fst_image_Sigma
tff(fact_5460_Restr__trancl__mono,axiom,
    ! [A: $tType,V2: A,W2: A,E6: set(product_prod(A,A)),U2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_trancl(A,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))),inf_inf(set(product_prod(A,A))),E6),product_Sigma(A,A,U2,aTP_Lamp_abo(set(A),fun(A,set(A)),U2))))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_trancl(A,E6))) ) ).

% Restr_trancl_mono
tff(fact_5461_butlast__subset,axiom,
    ! [A: $tType,Xs: list(A),A6: set(A)] :
      ( ( Xs != nil(A) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),butlast(A,Xs))),A6)) ) ) ).

% butlast_subset
tff(fact_5462_butlast__eq__consE,axiom,
    ! [A: $tType,L: list(A),X: A,Xs: list(A)] :
      ( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
     => ~ ! [Xl: A] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl),nil(A)))) ) ).

% butlast_eq_consE
tff(fact_5463_butlast__eq__cons__conv,axiom,
    ! [A: $tType,L: list(A),X: A,Xs: list(A)] :
      ( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
    <=> ? [Xl2: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl2),nil(A)))) ) ).

% butlast_eq_cons_conv
tff(fact_5464_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_5465_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E6: fun(C,set(A)),F4: fun(D,set(B)),A6: set(C),B5: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),image2(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_abx(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E6),F4))),product_Sigma(C,D,A6,aTP_Lamp_abt(set(D),fun(C,set(D)),B5)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),E6),A6)),aa(set(D),fun(A,set(B)),aTP_Lamp_aby(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B5)) ).

% UN_Times_distrib
tff(fact_5466_set__zip__cart,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),L: list(A),L4: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,L,L4))))
     => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_abd(list(B),fun(A,set(B)),L4)))) ) ).

% set_zip_cart
tff(fact_5467_nth__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),butlast(A,Xs))))
     => ( aa(nat,A,nth(A,butlast(A,Xs)),N) = aa(nat,A,nth(A,Xs),N) ) ) ).

% nth_butlast
tff(fact_5468_take__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,N,butlast(A,Xs)) = take(A,N,Xs) ) ) ).

% take_butlast
tff(fact_5469_butlast__upt,axiom,
    ! [M: nat,N: nat] : butlast(nat,upt(M,N)) = upt(M,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).

% butlast_upt
tff(fact_5470_butlast__rev__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( butlast(A,rev(A,Xs)) = rev(A,aa(list(A),list(A),tl(A),Xs)) ) ) ).

% butlast_rev_tl
tff(fact_5471_distinct__butlast__swap,axiom,
    ! [A: $tType,Pq: list(A),I2: nat] :
      ( distinct(A,Pq)
     => distinct(A,butlast(A,list_update(A,Pq,I2,last(A,Pq)))) ) ).

% distinct_butlast_swap
tff(fact_5472_sum_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,fun(C,A)),B5: set(C),A6: set(B)] : groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aTP_Lamp_fw(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B5),A6) = groups7311177749621191930dd_sum(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3),product_Sigma(B,C,A6,aTP_Lamp_abq(set(C),fun(B,set(C)),B5))) ) ).

% sum.cartesian_product
tff(fact_5473_prod_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,fun(C,A)),B5: set(C),A6: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_ig(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G3),B5),A6) = groups7121269368397514597t_prod(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3),product_Sigma(B,C,A6,aTP_Lamp_abq(set(C),fun(B,set(C)),B5))) ) ).

% prod.cartesian_product
tff(fact_5474_takeWhile__not__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( takeWhile(A,aTP_Lamp_abz(list(A),fun(A,bool),Xs),Xs) = butlast(A,Xs) ) ) ).

% takeWhile_not_last
tff(fact_5475_snd__image__Sigma,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B5: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A6,B5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6)) ).

% snd_image_Sigma
tff(fact_5476_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A6),aTP_Lamp_aca(set(product_prod(A,B)),fun(A,set(B)),A6)))) ).

% subset_fst_snd
tff(fact_5477_snoc__eq__iff__butlast_H,axiom,
    ! [A: $tType,Ys2: list(A),Xs: list(A),X: A] :
      ( ( Ys2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) )
    <=> ( ( Ys2 != nil(A) )
        & ( butlast(A,Ys2) = Xs )
        & ( last(A,Ys2) = X ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_5478_hd__butlast,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_5479_sum_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),B5: fun(B,set(C)),G3: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A6))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B5,X3))) )
           => ( groups7311177749621191930dd_sum(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_acb(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B5),G3),A6) = groups7311177749621191930dd_sum(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3),product_Sigma(B,C,A6,B5)) ) ) ) ) ).

% sum.Sigma
tff(fact_5480_prod_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B5: fun(B,set(C)),G3: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A6))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B5,X3))) )
           => ( groups7121269368397514597t_prod(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_acc(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B5),G3),A6) = groups7121269368397514597t_prod(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G3),product_Sigma(B,C,A6,B5)) ) ) ) ) ).

% prod.Sigma
tff(fact_5481_rel__fun__Collect__case__prodD,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool)),F3: fun(A,C),G3: fun(B,D),X6: set(product_prod(A,B)),X: product_prod(A,B)] :
      ( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A6,B5),F3),G3))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),X6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),X6))
         => pp(aa(D,bool,aa(C,fun(D,bool),B5,aa(product_prod(A,B),C,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)),X)),aa(product_prod(A,B),D,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),D),comp(B,D,product_prod(A,B),G3),product_snd(A,B)),X))) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_5482_relImage__relInvImage,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F3),A6),aa(set(B),fun(A,set(A)),aTP_Lamp_acd(fun(B,A),fun(set(B),fun(A,set(A))),F3),A6))))
     => ( bNF_Gr4221423524335903396lImage(B,A,bNF_Gr7122648621184425601vImage(B,A,A6,R2,F3),F3) = R2 ) ) ).

% relImage_relInvImage
tff(fact_5483_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: fun(A,set(B))] : product_Sigma(A,B,A6,B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_acf(fun(A,set(B)),fun(A,set(product_prod(A,B))),B5)),A6)) ).

% Sigma_def
tff(fact_5484_product__fold,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => ( product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_ach(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B5),bot_bot(set(product_prod(A,B))),A6) ) ) ) ).

% product_fold
tff(fact_5485_butlast__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( butlast(A,take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_5486_lists__length__Suc__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] : aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_aci(set(A),fun(nat,fun(list(A),bool)),A6),N)) = aa(set(product_prod(list(A),A)),set(list(A)),image2(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_so(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_mw(set(A),fun(nat,fun(list(A),bool)),A6),N)),aTP_Lamp_acj(set(A),fun(list(A),set(A)),A6))) ).

% lists_length_Suc_eq
tff(fact_5487_take__minus__one__conv__butlast,axiom,
    ! [A: $tType,N: nat,L: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L)))
     => ( take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,take(A,N,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_5488_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(nat,B),bool,aa(fun(nat,A),fun(fun(nat,B),bool),bNF_rel_fun(nat,nat,A,B,fequal(nat),R2),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_5489_Restr__subset,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( 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))),inf_inf(set(product_prod(A,A))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5)))),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))) = 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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))) ) ) ).

% Restr_subset
tff(fact_5490_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => pp(aa(fun(bool,B),bool,aa(fun(bool,A),fun(fun(bool,B),bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),R2),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).

% transfer_rule_of_bool
tff(fact_5491_fun__mono,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,C4: fun(A,fun(B,bool)),A6: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool)),D5: fun(C,fun(D,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),C4),A6))
     => ( pp(aa(fun(C,fun(D,bool)),bool,aa(fun(C,fun(D,bool)),fun(fun(C,fun(D,bool)),bool),ord_less_eq(fun(C,fun(D,bool))),B5),D5))
       => pp(aa(fun(fun(A,C),fun(fun(B,D),bool)),bool,aa(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(fun(A,C),fun(fun(B,D),bool)),bool),ord_less_eq(fun(fun(A,C),fun(fun(B,D),bool))),bNF_rel_fun(A,B,C,D,A6,B5)),bNF_rel_fun(A,B,C,D,C4,D5))) ) ) ).

% fun_mono
tff(fact_5492_pairself__image__cart,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B),B5: set(B)] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F3)),product_Sigma(B,B,A6,aTP_Lamp_ack(set(B),fun(B,set(B)),B5))) = product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F3),A6),aa(set(B),fun(A,set(A)),aTP_Lamp_acd(fun(B,A),fun(set(B),fun(A,set(A))),F3),B5)) ).

% pairself_image_cart
tff(fact_5493_plus__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_acl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).

% plus_rat.transfer
tff(fact_5494_pairself_Osimps,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: A,B2: A] : aa(product_prod(A,A),product_prod(B,B),pairself(A,B,F3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,A3)),aa(A,B,F3,B2)) ).

% pairself.simps
tff(fact_5495_pairself_Oelims,axiom,
    ! [B: $tType,A: $tType,X: fun(A,B),Xa2: product_prod(A,A),Y: product_prod(B,B)] :
      ( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X),Xa2) = Y )
     => ~ ! [A5: A,B4: A] :
            ( ( Xa2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4) )
           => ( Y != aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X,A5)),aa(A,B,X,B4)) ) ) ) ).

% pairself.elims
tff(fact_5496_Fract_Otransfer,axiom,
    pp(aa(fun(int,fun(int,rat)),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_acm(int,fun(int,product_prod(int,int)))),fract)) ).

% Fract.transfer
tff(fact_5497_uminus__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_acn(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat))) ).

% uminus_rat.transfer
tff(fact_5498_times__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_aco(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).

% times_rat.transfer
tff(fact_5499_inverse__rat_Otransfer,axiom,
    pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_acp(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).

% inverse_rat.transfer
tff(fact_5500_pairself_Opelims,axiom,
    ! [B: $tType,A: $tType,X: fun(A,B),Xa2: product_prod(A,A),Y: product_prod(B,B)] :
      ( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X),Xa2) = Y )
     => ( pp(aa(product_prod(fun(A,B),product_prod(A,A)),bool,accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B)),aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),X),Xa2)))
       => ~ ! [A5: A,B4: A] :
              ( ( Xa2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4) )
             => ( ( Y = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X,A5)),aa(A,B,X,B4)) )
               => ~ pp(aa(product_prod(fun(A,B),product_prod(A,A)),bool,accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B)),aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),X),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)))) ) ) ) ) ).

% pairself.pelims
tff(fact_5501_positive_Otransfer,axiom,
    pp(aa(fun(rat,bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(rat,bool),bool),bNF_rel_fun(product_prod(int,int),rat,bool,bool,pcr_rat,fequal(bool)),aTP_Lamp_acq(product_prod(int,int),bool)),positive)) ).

% positive.transfer
tff(fact_5502_times__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nn(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).

% times_int.transfer
tff(fact_5503_positive__add,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,positive,X))
     => ( pp(aa(rat,bool,positive,Y))
       => pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),X),Y))) ) ) ).

% positive_add
tff(fact_5504_nat_Otransfer,axiom,
    pp(aa(fun(int,nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),bool),bNF_rel_fun(product_prod(nat,nat),int,nat,nat,pcr_int,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),nat2)) ).

% nat.transfer
tff(fact_5505_int__transfer,axiom,
    pp(aa(fun(nat,int),bool,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_acr(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).

% int_transfer
tff(fact_5506_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
    <=> pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y),X))) ) ).

% less_rat_def
tff(fact_5507_uminus__int_Otransfer,axiom,
    pp(aa(fun(int,int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),bool),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),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_np(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).

% uminus_int.transfer
tff(fact_5508_of__int_Otransfer,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(fun(int,A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(int,A),bool),bNF_rel_fun(product_prod(nat,nat),int,A,A,pcr_int,fequal(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_nq(nat,fun(nat,A)))),ring_1_of_int(A))) ) ).

% of_int.transfer
tff(fact_5509_positive__rat,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(rat,bool,positive,aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B2))) ) ).

% positive_rat
tff(fact_5510_less__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ns(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).

% less_int.transfer
tff(fact_5511_less__eq__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_nu(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).

% less_eq_int.transfer
tff(fact_5512_plus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).

% plus_int.transfer
tff(fact_5513_minus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ny(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).

% minus_int.transfer
tff(fact_5514_positive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa(rat,bool,positive,X))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,X))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X))))) ) ).

% positive.rep_eq
tff(fact_5515_map__to__set__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K2: A,V2: B] : map_to_set(A,B,fun_upd(A,option(B),M,K2,aa(B,option(B),some(B),V2))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(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),K2),V2)),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))),minus_minus(set(product_prod(A,B))),map_to_set(A,B,M)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_acs(A,fun(product_prod(A,B),bool),K2)))) ).

% map_to_set_upd
tff(fact_5516_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aat(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_5517_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),Q2: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_act(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),P2),Q2))))
      <=> ! [Y4: A] :
            ( pp(aa(A,bool,P2,Y4))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q2),Y4)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5518_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: B,B5: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_acv(fun(B,A),fun(B,fun(set(B),fun(A,bool))),F3),A3),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F3,A3)),aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_acw(fun(B,A),fun(set(B),fun(A,bool)),F3),B5))) ).

% eq_or_mem_image_simp
tff(fact_5519_rtrancl__reflcl__absorb,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : 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_rtrancl(A,R2)),id2(A)) = transitive_rtrancl(A,R2) ).

% rtrancl_reflcl_absorb
tff(fact_5520_rtrancl__reflcl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_rtrancl(A,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))),R2),id2(A))) = transitive_rtrancl(A,R2) ).

% rtrancl_reflcl
tff(fact_5521_Eps__Opt__eq__None,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ( eps_Opt(A,P2) = none(A) )
    <=> ~ ? [X_12: A] : pp(aa(A,bool,P2,X_12)) ) ).

% Eps_Opt_eq_None
tff(fact_5522_trancl__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_trancl(A,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))),R3),id2(A))) = transitive_rtrancl(A,R3) ).

% trancl_reflcl
tff(fact_5523_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P2: fun(B,fun(B,bool))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F3)),aa(fun(product_prod(B,B),bool),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,bool)),fun(product_prod(B,B),bool),product_case_prod(B,B,bool),P2))) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_acx(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),F3),P2)) ).

% pairself_image_eq
tff(fact_5524_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,transitive_rtrancl(A,R3))),transitive_rtrancl(list(A),listrel1(A,R3)))) ).

% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_5525_rtrancl__mono,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),transitive_rtrancl(A,S2))) ) ).

% rtrancl_mono
tff(fact_5526_rtrancl__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,R2)))
       => ( transitive_rtrancl(A,S) = transitive_rtrancl(A,R2) ) ) ) ).

% rtrancl_subset
tff(fact_5527_rtrancl__subset__rtrancl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),transitive_rtrancl(A,S2)))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),transitive_rtrancl(A,S2))) ) ).

% rtrancl_subset_rtrancl
tff(fact_5528_rtrancl__mono__rightI,axiom,
    ! [A: $tType,S: set(product_prod(A,A)),S4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),S4))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,S4))) ) ).

% rtrancl_mono_rightI
tff(fact_5529_rtrancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V: set(product_prod(A,A)),X: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U2),V))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,U2)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,V))) ) ) ).

% rtrancl_mono_mp
tff(fact_5530_r__le__rtrancl,axiom,
    ! [A: $tType,S: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,S))) ).

% r_le_rtrancl
tff(fact_5531_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P2: fun(B,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_acy(fun(B,A),fun(fun(B,bool),fun(A,bool)),F3),P2)) = aa(set(B),set(A),image2(B,A,F3),aa(fun(B,bool),set(B),collect(B),P2)) ).

% setcompr_eq_image
tff(fact_5532_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_acw(fun(B,A),fun(set(B),fun(A,bool)),F3),A6)) = aa(set(B),set(A),image2(B,A,F3),A6) ).

% Setcompr_eq_image
tff(fact_5533_fs__contract,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,fun(B,C)),S: set(C)] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_acz(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),F3),S))) = aa(fun(A,bool),set(A),collect(A),aa(set(C),fun(A,bool),aTP_Lamp_ada(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),F3),S)) ).

% fs_contract
tff(fact_5534_tranclD,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ? [Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),R2))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),transitive_rtrancl(A,R2))) ) ) ).

% tranclD
tff(fact_5535_rtranclD,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2)))
     => ( ( A3 = B2 )
        | ( ( A3 != B2 )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))) ) ) ) ).

% rtranclD
tff(fact_5536_tranclD2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ? [Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),transitive_rtrancl(A,R2)))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y)),R2)) ) ) ).

% tranclD2
tff(fact_5537_trancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3))) ) ).

% trancl_into_rtrancl
tff(fact_5538_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)))
    <=> ( ( X = Y )
        | ( ( X != Y )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2))) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_5539_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R3))) ) ) ).

% rtrancl_into_trancl1
tff(fact_5540_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R3))) ) ) ).

% rtrancl_into_trancl2
tff(fact_5541_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Z4: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),transitive_trancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),transitive_trancl(A,R3))) ) ) ).

% rtrancl_trancl_trancl
tff(fact_5542_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R3))) ) ) ).

% trancl_rtrancl_trancl
tff(fact_5543_in__rtrancl__insert,axiom,
    ! [A: $tType,X: product_prod(A,A),R2: set(product_prod(A,A)),R3: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,R2)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),R3),R2)))) ) ).

% in_rtrancl_insert
tff(fact_5544_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Ys2: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),transitive_rtrancl(list(A),listrel1(A,R3))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))),transitive_rtrancl(list(A),listrel1(A,R3)))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_5545_set__Cons__def,axiom,
    ! [A: $tType,A6: set(A),XS: set(list(A))] : set_Cons(A,A6,XS) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),bool),aTP_Lamp_adb(set(A),fun(set(list(A)),fun(list(A),bool)),A6),XS)) ).

% set_Cons_def
tff(fact_5546_in__rtrancl__UnI,axiom,
    ! [A: $tType,X: product_prod(A,A),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,R2)))
        | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,S))) )
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X),transitive_rtrancl(A,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))),R2),S)))) ) ).

% in_rtrancl_UnI
tff(fact_5547_rtrancl__Un__rtrancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] : transitive_rtrancl(A,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_rtrancl(A,R2)),transitive_rtrancl(A,S))) = transitive_rtrancl(A,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))),R2),S)) ).

% rtrancl_Un_rtrancl
tff(fact_5548_converse__rtranclE_H,axiom,
    ! [A: $tType,U: A,V2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),transitive_rtrancl(A,R2)))
     => ( ( U != V2 )
       => ~ ! [Vh: A] :
              ( ( U != Vh )
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),Vh)),R2))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Vh),V2)),transitive_rtrancl(A,R2))) ) ) ) ) ).

% converse_rtranclE'
tff(fact_5549_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_rtrancl(A,R3))) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_5550_converse__rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(A,bool,P2,B2))
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_rtrancl(A,R3)))
               => ( pp(aa(A,bool,P2,Z3))
                 => pp(aa(A,bool,P2,Y3)) ) ) )
         => pp(aa(A,bool,P2,A3)) ) ) ) ).

% converse_rtrancl_induct
tff(fact_5551_converse__rtranclE,axiom,
    ! [A: $tType,X: A,Z4: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),transitive_rtrancl(A,R3)))
     => ( ( X != Z4 )
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R3))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4)),transitive_rtrancl(A,R3))) ) ) ) ).

% converse_rtranclE
tff(fact_5552_rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(A,bool,P2,A3))
       => ( ! [Y3: A,Z3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_rtrancl(A,R3)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
               => ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,P2,Z3)) ) ) )
         => pp(aa(A,bool,P2,B2)) ) ) ) ).

% rtrancl_induct
tff(fact_5553_rtrancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Z4: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),transitive_rtrancl(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),transitive_rtrancl(A,R3))) ) ) ).

% rtrancl_trans
tff(fact_5554_rtranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( ( A3 != B2 )
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_rtrancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2)),R3)) ) ) ) ).

% rtranclE
tff(fact_5555_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_rtrancl(A,R3))) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_5556_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A3: A,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),transitive_rtrancl(A,R3))) ).

% rtrancl.rtrancl_refl
tff(fact_5557_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R3)))
    <=> ( ? [A9: A] :
            ( ( A1 = A9 )
            & ( A22 = A9 ) )
        | ? [A9: A,B7: A,C5: A] :
            ( ( A1 = A9 )
            & ( A22 = C5 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),B7)),transitive_rtrancl(A,R3)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5)),R3)) ) ) ) ).

% rtrancl.simps
tff(fact_5558_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R3)))
     => ( ( A22 != A1 )
       => ~ ! [B4: A] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4)),transitive_rtrancl(A,R3)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22)),R3)) ) ) ) ).

% rtrancl.cases
tff(fact_5559_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(A,bool),Q2: fun(B,bool),F3: fun(A,fun(B,C))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),Q2)))
       => pp(aa(set(C),bool,finite_finite2(C),aa(fun(C,bool),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_adc(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),P2),Q2),F3)))) ) ) ).

% finite_image_set2
tff(fact_5560_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
     => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aTP_Lamp_add(fun(A,bool),fun(fun(A,B),fun(B,bool)),P2),F3)))) ) ).

% finite_image_set
tff(fact_5561_Union__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),P2: fun(B,bool)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(fun(B,bool),fun(set(A),bool),aTP_Lamp_ade(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),F3),P2))) = aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_adf(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),F3),P2)) ).

% Union_SetCompr_eq
tff(fact_5562_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A3: A,B2: A,P2: set(product_prod(A,A)),Q2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,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))),P2),Q2))))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),transitive_rtrancl(A,P2)))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),Q2))
             => ( X3 = Y3 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P2))) ) ) ).

% rtrancl_Un_separatorE
tff(fact_5563_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A3: A,B2: A,P2: set(product_prod(A,A)),Q2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,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))),P2),Q2))))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2)),transitive_rtrancl(A,P2)))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),Q2))
             => ( Y3 = X3 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P2))) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_5564_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R3)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By))
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R3))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R3)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay)) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_5565_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xa2: A,Xb2: B,Za: A,Zb: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R3)))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb2) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A5: A,B4: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),R3))
             => ~ pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R3))) ) ) ) ).

% converse_rtranclE2
tff(fact_5566_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R3)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay))
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),transitive_rtrancl(product_prod(A,B),R3)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R3))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).

% rtrancl_induct2
tff(fact_5567_rtrancl__Un__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),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_rtrancl(A,R2)),transitive_rtrancl(A,S))),transitive_rtrancl(A,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))),R2),S)))) ).

% rtrancl_Un_subset
tff(fact_5568_rtrancl__sub__insert__rtrancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),X),R2)))) ).

% rtrancl_sub_insert_rtrancl
tff(fact_5569_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_adg(fun(B,A),fun(A,bool),F3)) = aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_5570_finite__inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: A,A6: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_adh(A,fun(set(A),fun(A,bool)),A3),A6))) ) ).

% finite_inf_Sup
tff(fact_5571_Id__def,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_adi(product_prod(A,A),bool)) ).

% Id_def
tff(fact_5572_ran__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : ran(A,B,M) = aa(fun(B,bool),set(B),collect(B),aTP_Lamp_adj(fun(A,option(B)),fun(B,bool),M)) ).

% ran_def
tff(fact_5573_Pow__Compl,axiom,
    ! [A: $tType,A6: set(A)] : pow(A,aa(set(A),set(A),uminus_uminus(set(A)),A6)) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_adk(set(A),fun(set(A),bool),A6)) ).

% Pow_Compl
tff(fact_5574_listrel1__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : listrel1(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_adl(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% listrel1_def
tff(fact_5575_relImage__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F3: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R2,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_adm(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),R2),F3)) ).

% relImage_def
tff(fact_5576_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R2: set(product_prod(B,B)),F3: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A6,R2,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_adn(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),A6),R2),F3)) ).

% relInvImage_def
tff(fact_5577_trancl__union__outside,axiom,
    ! [A: $tType,V2: A,W2: A,E6: set(product_prod(A,A)),U2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_trancl(A,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))),E6),U2))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_trancl(A,E6)))
       => ? [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),X3)),transitive_rtrancl(A,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))),E6),U2))))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),U2))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),W2)),transitive_rtrancl(A,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))),E6),U2)))) ) ) ) ).

% trancl_union_outside
tff(fact_5578_trancl__over__edgeE,axiom,
    ! [A: $tType,U: A,W2: A,V1: A,V22: A,E6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W2)),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),V1),V22)),E6))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W2)),transitive_trancl(A,E6)))
       => ~ ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V1)),transitive_rtrancl(A,E6)))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V22),W2)),transitive_rtrancl(A,E6))) ) ) ) ).

% trancl_over_edgeE
tff(fact_5579_Un__interval,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [B1: A,B22: A,B32: A,F3: fun(A,B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B1),B22))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B22),B32))
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ado(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B22),F3))),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ado(A,fun(A,fun(fun(A,B),fun(B,bool))),B22),B32),F3))) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ado(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B32),F3)) ) ) ) ) ).

% Un_interval
tff(fact_5580_lex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_adp(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lex_conv
tff(fact_5581_rtrancl__trancl__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_rtrancl(A,R3) = 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,R3)),id2(A)) ).

% rtrancl_trancl_reflcl
tff(fact_5582_rtrancl__unfold,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : transitive_rtrancl(A,R3) = 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))),id2(A)),relcomp(A,A,A,transitive_rtrancl(A,R3),R3)) ).

% rtrancl_unfold
tff(fact_5583_Collect__ex__eq,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_adq(fun(A,fun(B,bool)),fun(A,bool),P2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ads(fun(A,fun(B,bool)),fun(B,set(A)),P2)),top_top(set(B)))) ).

% Collect_ex_eq
tff(fact_5584_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
       => ( ( A3 = B2 )
          | pp(aa(set(A),bool,member(A,A3),A6)) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_5585_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: set(product_prod(A,C)),S2: set(product_prod(C,B))] : relcomp(A,C,B,R3,S2) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_adt(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R3),S2))) ).

% relcomp_unfold
tff(fact_5586_Restr__rtrancl__mono,axiom,
    ! [A: $tType,V2: A,W2: A,E6: set(product_prod(A,A)),U2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_rtrancl(A,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))),inf_inf(set(product_prod(A,A))),E6),product_Sigma(A,A,U2,aTP_Lamp_abo(set(A),fun(A,set(A)),U2))))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_rtrancl(A,E6))) ) ).

% Restr_rtrancl_mono
tff(fact_5587_graph__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_adu(fun(A,option(B)),fun(product_prod(A,B),bool),M)) ).

% graph_def
tff(fact_5588_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N)),transitive_rtrancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_5589_rtrancl__mapI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: A,E6: set(product_prod(A,A)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,E6)))
     => pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,A3)),aa(A,B,F3,B2))),transitive_rtrancl(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),image2(product_prod(A,A),product_prod(B,B),pairself(A,B,F3)),E6)))) ) ).

% rtrancl_mapI
tff(fact_5590_set__map__filter,axiom,
    ! [A: $tType,B: $tType,G3: fun(B,option(A)),Xs: list(B)] : aa(list(A),set(A),set2(A),map_filter(B,A,G3,Xs)) = aa(fun(A,bool),set(A),collect(A),aa(list(B),fun(A,bool),aTP_Lamp_adv(fun(B,option(A)),fun(list(B),fun(A,bool)),G3),Xs)) ).

% set_map_filter
tff(fact_5591_ex__assn__def,axiom,
    ! [A: $tType,P2: fun(A,assn)] : ex_assn(A,P2) = abs_assn(aTP_Lamp_adw(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),bool),P2)) ).

% ex_assn_def
tff(fact_5592_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_adx(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_5593_rtrancl__Int__subset,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id2(A)),S2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,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))),inf_inf(set(product_prod(A,A))),transitive_rtrancl(A,R3)),S2),R3)),S2))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),S2)) ) ) ).

% rtrancl_Int_subset
tff(fact_5594_rtrancl__last__touch,axiom,
    ! [A: $tType,Q5: A,Q6: A,R2: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Q6)),transitive_rtrancl(A,R2)))
     => ( pp(aa(set(A),bool,member(A,Q5),S))
       => ~ ! [Qt: A] :
              ( pp(aa(set(A),bool,member(A,Qt),S))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Qt)),transitive_rtrancl(A,R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q6)),transitive_rtrancl(A,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))),minus_minus(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_abo(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).

% rtrancl_last_touch
tff(fact_5595_rtrancl__last__visit_H,axiom,
    ! [A: $tType,Q5: A,Q6: A,R2: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Q6)),transitive_rtrancl(A,R2)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Q6)),transitive_rtrancl(A,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))),minus_minus(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_abo(set(A),fun(A,set(A)),S))))))
       => ~ ! [Qt: A] :
              ( pp(aa(set(A),bool,member(A,Qt),S))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Qt)),transitive_rtrancl(A,R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q6)),transitive_rtrancl(A,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))),minus_minus(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_abo(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).

% rtrancl_last_visit'
tff(fact_5596_rtrancl__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),A3),B2)),R3)) = 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_rtrancl(A,R3)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_ady(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R3)))) ).

% rtrancl_insert
tff(fact_5597_rtrancl__imp__UN__relpow,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),transitive_rtrancl(A,R2)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P3),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),top_top(set(nat)))))) ) ).

% rtrancl_imp_UN_relpow
tff(fact_5598_rtrancl__is__UN__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),top_top(set(nat)))) ).

% rtrancl_is_UN_relpow
tff(fact_5599_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = aa(fun(A,bool),set(A),collect(A),aa(set(nat),fun(A,bool),aTP_Lamp_adz(list(A),fun(set(nat),fun(A,bool)),Xs),I5)) ).

% set_nths
tff(fact_5600_rtrancl__last__visit,axiom,
    ! [A: $tType,Q5: A,Q6: A,R2: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Q6)),transitive_rtrancl(A,R2)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Q6)),transitive_rtrancl(A,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))),minus_minus(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_abo(set(A),fun(A,set(A)),S))))))
       => ~ ! [Qt: A] :
              ( pp(aa(set(A),bool,member(A,Qt),S))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),Qt)),transitive_trancl(A,R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q6)),transitive_rtrancl(A,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))),minus_minus(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_abo(set(A),fun(A,set(A)),S)))))) ) ) ) ) ).

% rtrancl_last_visit
tff(fact_5601_trancl__insert,axiom,
    ! [A: $tType,Y: A,X: A,R3: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),Y),X)),R3)) = 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,R3)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_ady(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Y),X),R3)))) ).

% trancl_insert
tff(fact_5602_set__drop__conv,axiom,
    ! [A: $tType,N: nat,L: list(A)] : aa(list(A),set(A),set2(A),drop(A,N,L)) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_aea(nat,fun(list(A),fun(A,bool)),N),L)) ).

% set_drop_conv
tff(fact_5603_trancl__multi__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),M: A,X6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,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))),R3),product_Sigma(A,A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),M),bot_bot(set(A))),aTP_Lamp_abo(set(A),fun(A,set(A)),X6))))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
       => ~ ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),M)),transitive_rtrancl(A,R3)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2)),transitive_rtrancl(A,R3))) ) ) ) ) ).

% trancl_multi_insert2
tff(fact_5604_trancl__multi__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),X6: set(A),M: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,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))),R3),product_Sigma(A,A,X6,aTP_Lamp_aeb(A,fun(A,set(A)),M))))))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3)))
       => ~ ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),X6))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),transitive_rtrancl(A,R3)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),B2)),transitive_rtrancl(A,R3))) ) ) ) ) ).

% trancl_multi_insert
tff(fact_5605_rtrancl__last__visit__node,axiom,
    ! [A: $tType,S2: A,S3: A,R2: set(product_prod(A,A)),Sh: A] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S3)),transitive_rtrancl(A,R2)))
     => ( ( ( S2 != Sh )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),S3)),transitive_rtrancl(A,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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_aec(A,fun(A,set(A)),Sh)))))) )
        | ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S2),Sh)),transitive_rtrancl(A,R2)))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S3)),transitive_rtrancl(A,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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_aec(A,fun(A,set(A)),Sh)))))) ) ) ) ).

% rtrancl_last_visit_node
tff(fact_5606_brk__rel__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : brk_rel(A,B,R2) = aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B))),aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),fun(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B)))),sup_sup(set(product_prod(product_prod(bool,A),product_prod(bool,B)))),aa(fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),set(product_prod(product_prod(bool,A),product_prod(bool,B))),collect(product_prod(product_prod(bool,A),product_prod(bool,B))),aTP_Lamp_aed(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),R2))),aa(fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),set(product_prod(product_prod(bool,A),product_prod(bool,B))),collect(product_prod(product_prod(bool,A),product_prod(bool,B))),aTP_Lamp_aee(product_prod(product_prod(bool,A),product_prod(bool,B)),bool))) ).

% brk_rel_def
tff(fact_5607_lexn__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_aef(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R3),N))) ).

% lexn_conv
tff(fact_5608_rel__pred__comp__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),P2: fun(B,bool),X5: A] :
      ( rel_pred_comp(A,B,R2,P2,X5)
    <=> ? [Y4: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X5),Y4))
          & pp(aa(B,bool,P2,Y4)) ) ) ).

% rel_pred_comp_def
tff(fact_5609_lex__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = aa(set(set(product_prod(list(A),list(A)))),set(product_prod(list(A),list(A))),complete_Sup_Sup(set(product_prod(list(A),list(A)))),aa(set(nat),set(set(product_prod(list(A),list(A)))),image2(nat,set(product_prod(list(A),list(A))),lexn(A,R3)),top_top(set(nat)))) ).

% lex_def
tff(fact_5610_lexord__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lexord(A,R3) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_aeg(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lexord_def
tff(fact_5611_list__collect__set__alt,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F3,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),bool),aTP_Lamp_aeh(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F3),L))) ).

% list_collect_set_alt
tff(fact_5612_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),aa(nat,nat,suc,N)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),aa(set(product_prod(product_prod(A,list(A)),product_prod(A,list(A)))),set(product_prod(list(A),list(A))),image2(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A)),product_map_prod(product_prod(A,list(A)),list(A),product_prod(A,list(A)),list(A),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)))),lex_prod(A,list(A),R3,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N)))),aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_aei(nat,fun(list(A),fun(list(A),bool)),N)))) ).

% lexn.simps(2)
tff(fact_5613_map__prod__ident,axiom,
    ! [B: $tType,A: $tType,X5: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_cc(A,A),aTP_Lamp_aej(B,B)),X5) = X5 ).

% map_prod_ident
tff(fact_5614_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,A3)),aa(D,B,G3,B2)) ).

% map_prod_simp
tff(fact_5615_fst__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),X: product_prod(C,D)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),X)) = aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),X)) ).

% fst_map_prod
tff(fact_5616_snd__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,B),G3: fun(D,A),X: product_prod(C,D)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,D),product_prod(B,A),product_map_prod(C,B,D,A,F3,G3),X)) = aa(D,A,G3,aa(product_prod(C,D),D,product_snd(C,D),X)) ).

% snd_map_prod
tff(fact_5617_list__collect__set__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),A3: B] : list_collect_set(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),nil(B))) = aa(B,set(A),F3,A3) ).

% list_collect_set_simps(2)
tff(fact_5618_list__collect__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,set(A))] : list_collect_set(B,A,F3,nil(B)) = bot_bot(set(A)) ).

% list_collect_set_simps(1)
tff(fact_5619_list__collect__set__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),A3: B,L: list(B)] : list_collect_set(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),L)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F3,A3)),list_collect_set(B,A,F3,L)) ).

% list_collect_set_simps(3)
tff(fact_5620_list__collect__set__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B),L4: list(B)] : list_collect_set(B,A,F3,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F3,L)),list_collect_set(B,A,F3,L4)) ).

% list_collect_set_simps(4)
tff(fact_5621_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),F3: fun(A,C),G3: fun(B,D)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2))
     => pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F3,A3)),aa(B,D,G3,B2))),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F3,G3)),R2))) ) ).

% map_prod_imageI
tff(fact_5622_lexord__cons__cons,axiom,
    ! [A: $tType,A3: A,X: list(A),B2: A,Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R3)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
        | ( ( A3 = B2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,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)),X),Y)),lexord(A,R3))) ) ) ) ).

% lexord_cons_cons
tff(fact_5623_list__collect__set__map__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,set(A)),X: fun(C,B),A3: C] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),nil(C)))) = aa(B,set(A),F3,aa(C,B,X,A3)) ).

% list_collect_set_map_simps(2)
tff(fact_5624_list__collect__set__map__simps_I1_J,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,set(A)),X: fun(C,B)] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),nil(C))) = bot_bot(set(A)) ).

% list_collect_set_map_simps(1)
tff(fact_5625_list__collect__set__map__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,set(A)),X: fun(C,B),A3: C,L: list(C)] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),L))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F3,aa(C,B,X,A3))),list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),L))) ).

% list_collect_set_map_simps(3)
tff(fact_5626_list__collect__set__map__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,set(A)),X: fun(C,B),L: list(C),L4: list(C)] : list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L),L4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),L))),list_collect_set(B,A,F3,aa(list(C),list(B),map(C,B,X),L4))) ).

% list_collect_set_map_simps(4)
tff(fact_5627_fst__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,C),G3: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),comp(product_prod(C,D),C,product_prod(A,B),product_fst(C,D)),product_map_prod(A,C,B,D,F3,G3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_map_prod
tff(fact_5628_snd__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,D),G3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),comp(product_prod(D,C),C,product_prod(A,B),product_snd(D,C)),product_map_prod(A,D,B,C,F3,G3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),G3),product_snd(A,B)) ).

% snd_comp_map_prod
tff(fact_5629_map__prod__compose,axiom,
    ! [D: $tType,C: $tType,A: $tType,E: $tType,F2: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F2,D),G22: fun(B,F2)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),comp(E,C,A,F1),F22),aa(fun(B,F2),fun(B,D),comp(F2,D,B,G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F2)),fun(product_prod(A,B),product_prod(C,D)),comp(product_prod(E,F2),product_prod(C,D),product_prod(A,B),product_map_prod(E,C,F2,D,F1,G1)),product_map_prod(A,E,B,F2,F22,G22)) ).

% map_prod_compose
tff(fact_5630_map__prod_Ocompositionality,axiom,
    ! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: fun(C,E),G3: fun(D,F2),H: fun(A,C),I2: fun(B,D),Prod: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F2),product_map_prod(C,E,D,F2,F3,G3),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,H,I2),Prod)) = aa(product_prod(A,B),product_prod(E,F2),product_map_prod(A,E,B,F2,aa(fun(A,C),fun(A,E),comp(C,E,A,F3),H),aa(fun(B,D),fun(B,F2),comp(D,F2,B,G3),I2)),Prod) ).

% map_prod.compositionality
tff(fact_5631_map__prod_Ocomp,axiom,
    ! [A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,B: $tType,F3: fun(C,E),G3: fun(D,F2),H: fun(A,C),I2: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F2)),comp(product_prod(C,D),product_prod(E,F2),product_prod(A,B),product_map_prod(C,E,D,F2,F3,G3)),product_map_prod(A,C,B,D,H,I2)) = product_map_prod(A,E,B,F2,aa(fun(A,C),fun(A,E),comp(C,E,A,F3),H),aa(fun(B,D),fun(B,F2),comp(D,F2,B,G3),I2)) ).

% map_prod.comp
tff(fact_5632_prod_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T5: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_cc(A,A),aTP_Lamp_aej(B,B)),T5) = T5 ).

% prod.map_ident
tff(fact_5633_case__prod__map__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: fun(B,fun(C,A)),F3: fun(D,B),G3: fun(E,C),X: product_prod(D,E)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H),aa(product_prod(D,E),product_prod(B,C),product_map_prod(D,B,E,C,F3,G3),X)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_case_prod(D,E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aek(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),H),F3),G3)),X) ).

% case_prod_map_prod
tff(fact_5634_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod(A,B),F3: fun(C,A),G3: fun(D,B),R2: set(product_prod(C,D))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),C3),aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3)),R2)))
     => ~ ! [X3: C,Y3: D] :
            ( ( C3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),aa(D,B,G3,Y3)) )
           => ~ pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X3),Y3)),R2)) ) ) ).

% prod_fun_imageE
tff(fact_5635_lexord__irreflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lexord(A,R3))) ) ).

% lexord_irreflexive
tff(fact_5636_lexord__linear,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: list(A),Y: list(A)] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)),R3))
          | ( A5 = B4 )
          | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R3)) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,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)),X),Y)),lexord(A,R3)))
        | ( X = Y )
        | pp(aa(set(product_prod(list(A),list(A))),bool,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)),Y),X)),lexord(A,R3))) ) ) ).

% lexord_linear
tff(fact_5637_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(A,C),G3: fun(B,D)] : product_map_prod(A,C,B,D,F3,G3) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_ael(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F3),G3)) ).

% map_prod_def
tff(fact_5638_case__prod__o__map__prod,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F3: fun(D,fun(E,C)),G1: fun(A,D),G22: fun(B,E)] : aa(fun(product_prod(A,B),product_prod(D,E)),fun(product_prod(A,B),C),comp(product_prod(D,E),C,product_prod(A,B),aa(fun(D,fun(E,C)),fun(product_prod(D,E),C),product_case_prod(D,E,C),F3)),product_map_prod(A,D,B,E,G1,G22)) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_aem(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),F3),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_5639_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A)),Ys2: list(A),Zs: list(A)] :
      ( ! [X3: A,Y3: A,Z3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3)),R3)) ) ) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),lexord(A,R3)))
       => ( pp(aa(set(product_prod(list(A),list(A))),bool,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)),Ys2),Zs)),lexord(A,R3)))
         => pp(aa(set(product_prod(list(A),list(A))),bool,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),Zs)),lexord(A,R3))) ) ) ) ).

% lexord_partial_trans
tff(fact_5640_lexord__append__leftD,axiom,
    ! [A: $tType,X: list(A),U: list(A),V2: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V2))),lexord(A,R3)))
     => ( ! [A5: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),R3))
       => pp(aa(set(product_prod(list(A),list(A))),bool,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)),U),V2)),lexord(A,R3))) ) ) ).

% lexord_append_leftD
tff(fact_5641_list__collect__set__as__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F3,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(B),list(set(A)),map(B,set(A),F3),L))) ).

% list_collect_set_as_map
tff(fact_5642_map__prod__surj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(B,A),A6: set(B),A13: set(A),G3: fun(D,C),B5: set(D),B13: set(C)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A6) = A13 )
     => ( ( aa(set(D),set(C),image2(D,C,G3),B5) = B13 )
       => ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F3,G3)),product_Sigma(B,D,A6,aTP_Lamp_aen(set(D),fun(B,set(D)),B5))) = product_Sigma(A,C,A13,aTP_Lamp_abr(set(C),fun(A,set(C)),B13)) ) ) ) ).

% map_prod_surj_on
tff(fact_5643_map__prod__surj,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(A,B),G3: fun(C,D)] :
      ( ( aa(set(A),set(B),image2(A,B,F3),top_top(set(A))) = top_top(set(B)) )
     => ( ( aa(set(C),set(D),image2(C,D,G3),top_top(set(C))) = top_top(set(D)) )
       => ( aa(set(product_prod(A,C)),set(product_prod(B,D)),image2(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F3,G3)),top_top(set(product_prod(A,C)))) = top_top(set(product_prod(B,D))) ) ) ) ).

% map_prod_surj
tff(fact_5644_lexord__append__left__rightI,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
     => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)))),lexord(A,R3))) ) ).

% lexord_append_left_rightI
tff(fact_5645_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R3)))
    <=> ( ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R3)) )
        | pp(aa(set(product_prod(list(A),list(A))),bool,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)),Ys2),Zs)),lexord(A,R3))) ) ) ).

% lexord_same_pref_iff
tff(fact_5646_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R3: set(product_prod(A,A)),V2: list(A),Z4: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,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)),U),W2)),lexord(A,R3)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z4))),lexord(A,R3))) ) ) ).

% lexord_sufI
tff(fact_5647_list__collect__set__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F3,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),bool),aTP_Lamp_aeo(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F3),L))) ).

% list_collect_set_def
tff(fact_5648_List_Olexordp__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys2: list(A)] :
      ( lexordp(A,R3,Xs,Ys2)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% List.lexordp_def
tff(fact_5649_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A6: set(C),F3: fun(C,A),G3: fun(C,B)] : bNF_Greatest_image2(C,A,B,A6,F3,G3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_aep(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A6),F3),G3)) ).

% image2_def
tff(fact_5650_of__rat_Otransfer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(fun(rat,A),bool,aa(fun(product_prod(int,int),A),fun(fun(rat,A),bool),bNF_rel_fun(product_prod(int,int),rat,A,A,pcr_rat,fequal(A)),aTP_Lamp_aeq(product_prod(int,int),A)),field_char_0_of_rat(A))) ) ).

% of_rat.transfer
tff(fact_5651_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3)) ) ) ).

% zero_less_of_rat_iff
tff(fact_5652_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),zero_zero(rat))) ) ) ).

% of_rat_less_0_iff
tff(fact_5653_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R3)) ) ) ).

% zero_le_of_rat_iff
tff(fact_5654_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),zero_zero(rat))) ) ) ).

% of_rat_le_0_iff
tff(fact_5655_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R3)) ) ) ).

% one_less_of_rat_iff
tff(fact_5656_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),one_one(rat))) ) ) ).

% of_rat_less_1_iff
tff(fact_5657_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R3)) ) ) ).

% one_le_of_rat_iff
tff(fact_5658_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),one_one(rat))) ) ) ).

% of_rat_le_1_iff
tff(fact_5659_of__rat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F3: fun(B,rat),A6: set(B)] : aa(rat,A,field_char_0_of_rat(A),groups7311177749621191930dd_sum(B,rat,F3,A6)) = groups7311177749621191930dd_sum(B,A,aTP_Lamp_aer(fun(B,rat),fun(B,A),F3),A6) ) ).

% of_rat_sum
tff(fact_5660_of__rat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F3: fun(B,rat),A6: set(B)] : aa(rat,A,field_char_0_of_rat(A),groups7121269368397514597t_prod(B,rat,F3,A6)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_aer(fun(B,rat),fun(B,A),F3),A6) ) ).

% of_rat_prod
tff(fact_5661_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat,S2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),aa(rat,A,field_char_0_of_rat(A),S2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),S2)) ) ) ).

% of_rat_less
tff(fact_5662_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat,S2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),aa(rat,A,field_char_0_of_rat(A),S2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),S2)) ) ) ).

% of_rat_less_eq
tff(fact_5663_of__rat__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_add
tff(fact_5664_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F3: fun(B,A),X: B,C3: C,G3: fun(B,C),A6: set(B)] :
      ( ( B2 = aa(B,A,F3,X) )
     => ( ( C3 = aa(B,C,G3,X) )
       => ( pp(aa(set(B),bool,member(B,X),A6))
         => pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C3)),bNF_Greatest_image2(B,A,C,A6,F3,G3))) ) ) ) ).

% image2_eqI
tff(fact_5665_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_aes(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys2)) ).

% set_zip
tff(fact_5666_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,B),B5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A6,F3)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),B5)) ) ).

% Gr_incl
tff(fact_5667_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_aet(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_5668_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% min.bounded_iff
tff(fact_5669_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5670_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb1
tff(fact_5671_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N)))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N) = M ) ) ) ).

% min_arg_le(2)
tff(fact_5672_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N: A,M: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N)))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N) = N ) ) ) ).

% min_arg_le(1)
tff(fact_5673_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N) = N )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),M)) ) ) ).

% min_eq_arg(2)
tff(fact_5674_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N) = M )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),N)) ) ) ).

% min_eq_arg(1)
tff(fact_5675_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z4: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),Y)) ) ) ) ).

% min_less_iff_conj
tff(fact_5676_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_5677_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb3
tff(fact_5678_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min_simps(2)
tff(fact_5679_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min_simps(1)
tff(fact_5680_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% min_less_self_conv(2)
tff(fact_5681_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% min_less_self_conv(1)
tff(fact_5682_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N)),N))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N) = N ) ) ) ).

% min_arg_not_ge(2)
tff(fact_5683_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),N)),M))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),N) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_5684_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_5685_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).

% min_bot
tff(fact_5686_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).

% min_top
tff(fact_5687_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).

% min_top2
tff(fact_5688_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = Y ) ).

% max_min_same(4)
tff(fact_5689_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_5690_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),X) = X ) ).

% max_min_same(2)
tff(fact_5691_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = X ) ).

% max_min_same(1)
tff(fact_5692_min__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_5693_min__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% min_0L
tff(fact_5694_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ).

% min_Suc_Suc
tff(fact_5695_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(1)
tff(fact_5696_min__Suc__gt_I1_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A3)),B2) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(1)
tff(fact_5697_min__Suc__gt_I2_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),aa(nat,nat,suc,A3)) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(2)
tff(fact_5698_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(2)
tff(fact_5699_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = aa(num,A,numeral_numeral(A),V2) ) ) ) ) ).

% min_number_of(3)
tff(fact_5700_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)) ) ) ) ) ).

% min_number_of(4)
tff(fact_5701_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I2: nat,X: A,J2: nat,Y: B] : zip(A,B,replicate(A,I2,X),replicate(B,J2,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),J2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_replicate
tff(fact_5702_GrD2,axiom,
    ! [A: $tType,B: $tType,X: A,Fx: B,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A6,F3)))
     => ( aa(A,B,F3,X) = Fx ) ) ).

% GrD2
tff(fact_5703_GrD1,axiom,
    ! [B: $tType,A: $tType,X: A,Fx: B,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A6,F3)))
     => pp(aa(set(A),bool,member(A,X),A6)) ) ).

% GrD1
tff(fact_5704_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa) = X5 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa) = Xa ) ) ) ) ).

% min_def_raw
tff(fact_5705_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z4))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z4)) ) ) ) ).

% min_le_iff_disj
tff(fact_5706_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).

% min.coboundedI2
tff(fact_5707_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).

% min.coboundedI1
tff(fact_5708_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_5709_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_5710_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2)) ) ).

% min.cobounded2
tff(fact_5711_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3)) ) ).

% min.cobounded1
tff(fact_5712_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.order_iff
tff(fact_5713_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))) ) ) ) ).

% min.boundedI
tff(fact_5714_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% min.boundedE
tff(fact_5715_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% min.orderI
tff(fact_5716_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.orderE
tff(fact_5717_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),D3))) ) ) ) ).

% min.mono
tff(fact_5718_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_5719_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).

% min_absorb1
tff(fact_5720_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ) ).

% min_def
tff(fact_5721_min__diff,axiom,
    ! [M: nat,I2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),I2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),I2) ).

% min_diff
tff(fact_5722_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).

% min.strict_coboundedI2
tff(fact_5723_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).

% min.strict_coboundedI1
tff(fact_5724_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
            & ( A3 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_5725_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% min.strict_boundedE
tff(fact_5726_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z4))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z4)) ) ) ) ).

% min_less_iff_disj
tff(fact_5727_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_5728_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)) ).

% nat_mult_min_right
tff(fact_5729_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) ).

% nat_mult_min_left
tff(fact_5730_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z4) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z4)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z4)) ) ).

% min_add_distrib_left
tff(fact_5731_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),Y),Z4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z4)) ) ).

% min_add_distrib_right
tff(fact_5732_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5733_Min_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% Min.semilattice_order_set_axioms
tff(fact_5734_le__prod__encode__2,axiom,
    ! [B2: nat,A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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),A3),B2)))) ).

% le_prod_encode_2
tff(fact_5735_le__prod__encode__1,axiom,
    ! [A3: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),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),A3),B2)))) ).

% le_prod_encode_1
tff(fact_5736_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_5737_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_5738_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_5739_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_5740_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_5741_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_5742_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( ( S = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = X ) )
            & ( ( S != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,complete_Inf_Inf(A),S)) ) ) ) ) ) ).

% Inf_insert_finite
tff(fact_5743_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_aeu(nat,fun(nat,nat),N),M) ).

% min_Suc1
tff(fact_5744_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_aev(nat,fun(nat,nat),N),M) ).

% min_Suc2
tff(fact_5745_Gr__def,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] : bNF_Gr(A,B,A6,F3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_aew(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A6),F3)) ).

% Gr_def
tff(fact_5746_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K2,M)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K2)),M) ).

% prod_encode_prod_decode_aux
tff(fact_5747_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,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)),X),Y)),lexord(A,R3)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
          & ( take(A,aa(list(A),nat,size_size(list(A)),X),Y) = X ) )
        | ? [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))))
            & ( take(A,I,X) = take(A,I,Y) )
            & pp(aa(set(product_prod(A,A)),bool,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,X),I)),aa(nat,A,nth(A,Y),I))),R3)) ) ) ) ).

% lexord_take_index_conv
tff(fact_5748_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),A,case_list(A,A,X,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_aex(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_5749_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F3: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aey(fun(B,list(A)),fun(B,set(A)),F3)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_5750_rtrancl__restrictI,axiom,
    ! [A: $tType,U: A,V2: A,E6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),transitive_rtrancl(A,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))),minus_minus(set(product_prod(A,A))),E6),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_abo(set(A),fun(A,set(A)),R2))))))
     => ( ~ pp(aa(set(A),bool,member(A,U),R2))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),transitive_rtrancl(A,rel_restrict(A,E6,R2)))) ) ) ).

% rtrancl_restrictI
tff(fact_5751_rel__restrict__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : rel_restrict(A,R2,bot_bot(set(A))) = R2 ).

% rel_restrict_empty
tff(fact_5752_finite__rel__restrict,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),rel_restrict(A,R2,A6))) ) ).

% finite_rel_restrict
tff(fact_5753_rel__restrict__mono,axiom,
    ! [A: $tType,A6: set(product_prod(A,A)),B5: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A6),B5))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A6,R2)),rel_restrict(A,B5,R2))) ) ).

% rel_restrict_mono
tff(fact_5754_rel__restrict__sub,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R2,A6)),R2)) ).

% rel_restrict_sub
tff(fact_5755_rel__restrict__union,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A),B5: set(A)] : rel_restrict(A,R2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = rel_restrict(A,rel_restrict(A,R2,A6),B5) ).

% rel_restrict_union
tff(fact_5756_rel__restrict__notR_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,A6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,A6,R2)))
     => ~ pp(aa(set(A),bool,member(A,Y),R2)) ) ).

% rel_restrict_notR(2)
tff(fact_5757_rel__restrict__notR_I1_J,axiom,
    ! [A: $tType,X: A,Y: A,A6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,A6,R2)))
     => ~ pp(aa(set(A),bool,member(A,X),R2)) ) ).

% rel_restrict_notR(1)
tff(fact_5758_rel__restrictI,axiom,
    ! [A: $tType,X: A,R2: set(A),Y: A,E6: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,member(A,X),R2))
     => ( ~ pp(aa(set(A),bool,member(A,Y),R2))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),E6))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,E6,R2))) ) ) ) ).

% rel_restrictI
tff(fact_5759_rel__restrict__lift,axiom,
    ! [A: $tType,X: A,Y: A,E6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),rel_restrict(A,E6,R2)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),E6)) ) ).

% rel_restrict_lift
tff(fact_5760_rel__restrict__trancl__mem,axiom,
    ! [A: $tType,A3: A,B2: A,A6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,rel_restrict(A,A6,R2))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),rel_restrict(A,transitive_trancl(A,A6),R2))) ) ).

% rel_restrict_trancl_mem
tff(fact_5761_rel__restrict__trancl__notR_I1_J,axiom,
    ! [A: $tType,V2: A,W2: A,E6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_trancl(A,rel_restrict(A,E6,R2))))
     => ~ pp(aa(set(A),bool,member(A,V2),R2)) ) ).

% rel_restrict_trancl_notR(1)
tff(fact_5762_rel__restrict__trancl__notR_I2_J,axiom,
    ! [A: $tType,V2: A,W2: A,E6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V2),W2)),transitive_trancl(A,rel_restrict(A,E6,R2))))
     => ~ pp(aa(set(A),bool,member(A,W2),R2)) ) ).

% rel_restrict_trancl_notR(2)
tff(fact_5763_rel__restrict__mono2,axiom,
    ! [A: $tType,R2: set(A),S: set(A),A6: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R2),S))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A6,S)),rel_restrict(A,A6,R2))) ) ).

% rel_restrict_mono2
tff(fact_5764_rel__restrict__trancl__sub,axiom,
    ! [A: $tType,A6: set(product_prod(A,A)),R2: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,rel_restrict(A,A6,R2))),rel_restrict(A,transitive_trancl(A,A6),R2))) ).

% rel_restrict_trancl_sub
tff(fact_5765_rel__restrict__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A)] : rel_restrict(A,R2,A6) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_aez(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),R2),A6))) ).

% rel_restrict_def
tff(fact_5766_rel__restrict__compl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A)] : 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))),inf_inf(set(product_prod(A,A))),rel_restrict(A,R2,A6)),rel_restrict(A,R2,aa(set(A),set(A),uminus_uminus(set(A)),A6))) = bot_bot(set(product_prod(A,A))) ).

% rel_restrict_compl
tff(fact_5767_homo__rel__restrict__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B5: set(A),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R2,A6)),product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6),aa(set(A),fun(A,set(A)),aTP_Lamp_afa(set(A),fun(set(A),fun(A,set(A))),B5),A6)))) ) ).

% homo_rel_restrict_mono
tff(fact_5768_rel__restrict__alt__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A)] : rel_restrict(A,R2,A6) = 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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(A),set(A),uminus_uminus(set(A)),A6),aTP_Lamp_afb(set(A),fun(A,set(A)),A6))) ).

% rel_restrict_alt_def
tff(fact_5769_rel__restrict__Sigma__sub,axiom,
    ! [A: $tType,A6: set(A),R2: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,transitive_trancl(A,product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))),R2)),transitive_trancl(A,product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),R2),aa(set(A),fun(A,set(A)),aTP_Lamp_afa(set(A),fun(set(A),fun(A,set(A))),A6),R2))))) ).

% rel_restrict_Sigma_sub
tff(fact_5770_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_aex(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) ) )
           => ~ ( ( X = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_5771_times__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nn(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nn(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% times_int.rsp
tff(fact_5772_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),A6: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_afc(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3)),A6) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F3),A6),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_afd(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F3),G3),A6)) ).

% image_split_eq_Sigma
tff(fact_5773_vimage__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),B5: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)))
    <=> pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),B5)) ) ).

% vimage_eq
tff(fact_5774_vimageI,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: B,B2: A,B5: set(A)] :
      ( ( aa(B,A,F3,A3) = B2 )
     => ( pp(aa(set(A),bool,member(A,B2),B5))
       => pp(aa(set(B),bool,member(B,A3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B5))) ) ) ).

% vimageI
tff(fact_5775_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),P2: fun(B,bool)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(fun(B,bool),set(B),collect(B),P2)) = aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_afe(fun(A,B),fun(fun(B,bool),fun(A,bool)),F3),P2)) ).

% vimage_Collect_eq
tff(fact_5776_vimage__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_cc(A,A)),Y6) = Y6 ).

% vimage_ident
tff(fact_5777_vimage__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),bot_bot(set(B))) = bot_bot(set(A)) ).

% vimage_empty
tff(fact_5778_vimage__UNIV,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),top_top(set(B))) = top_top(set(A)) ).

% vimage_UNIV
tff(fact_5779_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(B),B5: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)) ).

% vimage_Int
tff(fact_5780_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(B),B5: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)) ).

% vimage_Un
tff(fact_5781_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: nat] :
      ( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V2)))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_5782_vimage__const,axiom,
    ! [B: $tType,A: $tType,C3: B,A6: set(B)] :
      ( ( pp(aa(set(B),bool,member(B,C3),A6))
       => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aTP_Lamp_uf(B,fun(A,B),C3)),A6) = top_top(set(A)) ) )
      & ( ~ pp(aa(set(B),bool,member(B,C3),A6))
       => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aTP_Lamp_uf(B,fun(A,B),C3)),A6) = bot_bot(set(A)) ) ) ) ).

% vimage_const
tff(fact_5783_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(A)] : aa(set(B),set(A),image2(B,A,F3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(B),set(A),image2(B,A,F3),top_top(set(B)))) ).

% image_vimage_eq
tff(fact_5784_vimage__if,axiom,
    ! [B: $tType,A: $tType,C3: B,A6: set(B),D3: B,B5: set(A)] :
      ( ( pp(aa(set(B),bool,member(B,C3),A6))
       => ( ( pp(aa(set(B),bool,member(B,D3),A6))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aff(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B5)),A6) = top_top(set(A)) ) )
          & ( ~ pp(aa(set(B),bool,member(B,D3),A6))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aff(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B5)),A6) = B5 ) ) ) )
      & ( ~ pp(aa(set(B),bool,member(B,C3),A6))
       => ( ( pp(aa(set(B),bool,member(B,D3),A6))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aff(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B5)),A6) = aa(set(A),set(A),uminus_uminus(set(A)),B5) ) )
          & ( ~ pp(aa(set(B),bool,member(B,D3),A6))
           => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aff(B,fun(B,fun(set(A),fun(A,B))),C3),D3),B5)),A6) = bot_bot(set(A)) ) ) ) ) ) ).

% vimage_if
tff(fact_5785_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,B),G3: fun(A,B),Y: set(B)] :
      ( ! [W: A] :
          ( pp(aa(set(A),bool,member(A,W),S))
         => ( aa(A,B,F3,W) = aa(A,B,G3,W) ) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),Y)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),G3),Y)),S) ) ) ).

% vimage_inter_cong
tff(fact_5786_subset__vimage__iff,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,B),B5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A6))
         => pp(aa(set(B),bool,member(B,aa(A,B,F3,X4)),B5)) ) ) ).

% subset_vimage_iff
tff(fact_5787_vimage__mono,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(A),F3: fun(B,A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A6)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B5))) ) ).

% vimage_mono
tff(fact_5788_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(B),B5: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)) ).

% vimage_Diff
tff(fact_5789_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),uminus_uminus(set(B)),A6)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)) ).

% vimage_Compl
tff(fact_5790_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),B2: B] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B2),bot_bot(set(B))))))
    <=> ( aa(A,B,F3,A3) = B2 ) ) ).

% vimage_singleton_eq
tff(fact_5791_vimage__Collect,axiom,
    ! [B: $tType,A: $tType,P2: fun(B,bool),F3: fun(A,B),Q2: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(B,bool,P2,aa(A,B,F3,X3)))
        <=> pp(aa(A,bool,Q2,X3)) )
     => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(fun(B,bool),set(B),collect(B),P2)) = aa(fun(A,bool),set(A),collect(A),Q2) ) ) ).

% vimage_Collect
tff(fact_5792_vimageI2,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: B,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,aa(B,A,F3,A3)),A6))
     => pp(aa(set(B),bool,member(B,A3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A6))) ) ).

% vimageI2
tff(fact_5793_vimageE,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),B5: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)))
     => pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),B5)) ) ).

% vimageE
tff(fact_5794_vimageD,axiom,
    ! [A: $tType,B: $tType,A3: A,F3: fun(A,B),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)))
     => pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),A6)) ) ).

% vimageD
tff(fact_5795_vimage__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),B5: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_afg(fun(A,B),fun(set(B),fun(A,bool)),F3),B5)) ).

% vimage_def
tff(fact_5796_image__subset__iff__subset__vimage,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A6)),B5))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B5))) ) ).

% image_subset_iff_subset_vimage
tff(fact_5797_image__vimage__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),A6))),A6)) ).

% image_vimage_subset
tff(fact_5798_vimage__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),B5: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_afh(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F3),B5)),A6)) ).

% vimage_UN
tff(fact_5799_vimage__INT,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),B5: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A6))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_afh(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F3),B5)),A6)) ).

% vimage_INT
tff(fact_5800_vimage__Times,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,product_prod(B,C)),A6: set(B),B5: set(C)] : aa(set(product_prod(B,C)),set(A),aa(fun(A,product_prod(B,C)),fun(set(product_prod(B,C)),set(A)),vimage(A,product_prod(B,C)),F3),product_Sigma(B,C,A6,aTP_Lamp_abq(set(C),fun(B,set(C)),B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F3)),A6)),aa(set(C),set(A),aa(fun(A,C),fun(set(C),set(A)),vimage(A,C),aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F3)),B5)) ).

% vimage_Times
tff(fact_5801_vimage__Union,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(set(B))] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3)),A6)) ).

% vimage_Union
tff(fact_5802_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,X: B,A6: set(B),F3: fun(B,set(A))] :
      ( ( pp(aa(set(B),bool,member(B,X),A6))
       => ( aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X)),product_Sigma(B,A,A6,F3)) = aa(B,set(A),F3,X) ) )
      & ( ~ pp(aa(set(B),bool,member(B,X),A6))
       => ( aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X)),product_Sigma(B,A,A6,F3)) = bot_bot(set(A)) ) ) ) ).

% Pair_vimage_Sigma
tff(fact_5803_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),B5: set(A),A6: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),B5)),A6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F3),A6))) ) ) ).

% vimage_subsetD
tff(fact_5804_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A3: B,B5: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)) ).

% vimage_insert
tff(fact_5805_vimage__fst,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : aa(set(A),set(product_prod(A,B)),aa(fun(product_prod(A,B),A),fun(set(A),set(product_prod(A,B))),vimage(product_prod(A,B),A),product_fst(A,B)),A6) = product_Sigma(A,B,A6,aTP_Lamp_abb(A,set(B))) ).

% vimage_fst
tff(fact_5806_vimage__snd,axiom,
    ! [A: $tType,B: $tType,A6: set(B)] : aa(set(B),set(product_prod(A,B)),aa(fun(product_prod(A,B),B),fun(set(B),set(product_prod(A,B))),vimage(product_prod(A,B),B),product_snd(A,B)),A6) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aaz(set(B),fun(A,set(B)),A6)) ).

% vimage_snd
tff(fact_5807_uminus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),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_np(nat,fun(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_np(nat,fun(nat,product_prod(nat,nat)))))) ).

% uminus_int.rsp
tff(fact_5808_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_cc(A,A),Option) ).

% option.the_def
tff(fact_5809_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,case_list(A,A,undefined(A),aTP_Lamp_afi(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_5810_nat_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),nat,nat,intrel,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)))) ).

% nat.rsp
tff(fact_5811_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),aa(set(A),set(B),image2(A,B,F3),top_top(set(A)))))
       => pp(aa(set(B),bool,finite_finite2(B),A6)) ) ) ).

% finite_vimageD'
tff(fact_5812_of__int_Orsp,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(fun(product_prod(nat,nat),A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(product_prod(nat,nat),A),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),A,A,intrel,fequal(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_nq(nat,fun(nat,A)))),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_nq(nat,fun(nat,A))))) ) ).

% of_int.rsp
tff(fact_5813_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),B5: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_afj(fun(A,B),fun(B,set(A)),F3)),B5)) ).

% vimage_eq_UN
tff(fact_5814_intrel__def,axiom,
    intrel = aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_afl(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).

% intrel_def
tff(fact_5815_less__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ns(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ns(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_int.rsp
tff(fact_5816_less__eq__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_nu(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_nu(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_eq_int.rsp
tff(fact_5817_plus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% plus_int.rsp
tff(fact_5818_minus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ny(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ny(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% minus_int.rsp
tff(fact_5819_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),X))
           => ( ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_aex(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) )
                   => ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) )
             => ~ ( ( X = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),nil(A))) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_5820_mod__h__bot__normalize,axiom,
    ! [A: $tType,H: heap_ext(product_unit),P2: assn] :
      ( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H)
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))))
      <=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat))))) ) ) ).

% mod_h_bot_normalize
tff(fact_5821_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa2: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa2) = Y )
         => ( ! [X3: A] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( Y != X3 ) )
           => ( ! [X3: A,Y3: A,Zs2: list(A)] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                 => ( Y != if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))) ) )
             => ~ ( ( Xa2 = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_5822_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))),X,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) ) ).

% arg_min_list.simps(2)
tff(fact_5823_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa2: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa2) = Y )
         => ( pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),Xa2)))
           => ( ! [X3: A] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                 => ( ( Y = X3 )
                   => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) )
             => ( ! [X3: A,Y3: A,Zs2: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                   => ( ( Y = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))) )
                     => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))))) ) )
               => ~ ( ( Xa2 = nil(A) )
                   => ( ( Y = undefined(A) )
                     => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X),nil(A)))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_5824_add_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [X: A,Y: A,A3: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ) ) ).

% add.safe_commute
tff(fact_5825_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F3),top_top(set(nat)))))
         => ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
           => ( ! [N4: nat] :
                  ( ( aa(nat,A,F3,N4) = aa(nat,A,F3,aa(nat,nat,suc,N4)) )
                 => ( aa(nat,A,F3,aa(nat,nat,suc,N4)) = aa(nat,A,F3,aa(nat,nat,suc,aa(nat,nat,suc,N4))) ) )
             => ? [N11: nat] :
                  ( ! [N8: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N11))
                     => ! [M4: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N11))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N8))
                           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,M4)),aa(nat,A,F3,N8))) ) ) )
                  & ! [N8: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N11),N8))
                     => ( aa(nat,A,F3,N11) = aa(nat,A,F3,N8) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_5826_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F3: fun(A,B),A6: A,B5: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),B5))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,A6)),aa(A,B,F3,B5)))) ) ) ).

% mono_inf
tff(fact_5827_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F3: fun(A,B),A6: A,B5: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,A6)),aa(A,B,F3,B5))),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),B5)))) ) ) ).

% mono_sup
tff(fact_5828_mono__iff__le__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
        <=> ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N3)),aa(nat,A,F3,aa(nat,nat,suc,N3)))) ) ) ).

% mono_iff_le_Suc
tff(fact_5829_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
        <=> ! [X4: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y4))) ) ) ) ).

% mono_def
tff(fact_5830_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
         => pp(aa(fun(A,B),bool,order_mono(A,B),F3)) ) ) ).

% monoI
tff(fact_5831_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoE
tff(fact_5832_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoD
tff(fact_5833_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mono_invE
tff(fact_5834_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),A6: A,B5: A,N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),A6)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),B5))) ) ) ) ).

% funpow_mono
tff(fact_5835_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q2: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q2))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),aTP_Lamp_afm(fun(A,A),fun(nat,A),Q2))) ) ) ).

% mono_funpow
tff(fact_5836_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3))) ) ) ).

% mono_pow
tff(fact_5837_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mono_strict_invE
tff(fact_5838_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A] : pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),plus_plus(A),A3))) ) ).

% mono_add
tff(fact_5839_mono__Suc,axiom,
    pp(aa(fun(nat,nat),bool,order_mono(nat,nat),suc)) ).

% mono_Suc
tff(fact_5840_Rings_Omono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),times_times(A),A3))) ) ) ).

% Rings.mono_mult
tff(fact_5841_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),N))) ) ).

% mono_times_nat
tff(fact_5842_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F3: fun(A,B),M: A,N: A,M8: B,N2: B] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( ( aa(set(A),set(B),image2(A,B,F3),set_or7035219750837199246ssThan(A,M,N)) = set_or7035219750837199246ssThan(B,M8,N2) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
             => ( aa(A,B,F3,M) = M8 ) ) ) ) ) ).

% mono_image_least
tff(fact_5843_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F3: fun(A,A),P3: A,K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,P3)),P3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A))),P3)) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_5844_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F3: fun(A,A),P3: A,K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,F3,P3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),top_top(A)))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_5845_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),I2: nat,J2: nat,X: A,Y: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F3,X)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I2),F3),X)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J2),F3),Y))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_5846_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A6)))) ) ) ).

% mono_Sup
tff(fact_5847_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afn(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6)),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A6),I5))))) ) ) ).

% mono_SUP
tff(fact_5848_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A6),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afn(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6)),I5)))) ) ) ).

% mono_INF
tff(fact_5849_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A6))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A6)))) ) ) ).

% mono_Inf
tff(fact_5850_add_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.right_assoc
tff(fact_5851_add_Oright__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2) ) ).

% add.right_commute
tff(fact_5852_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,N: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),bot_bot(A)))) ) ) ) ).

% funpow_decreasing
tff(fact_5853_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,N: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_5854_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),K2))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aTP_Lamp_afo(nat,fun(nat,nat),K2))) ) ).

% mono_ge2_power_minus_self
tff(fact_5855_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys2 )
    <=> ? [F11: fun(nat,nat)] :
          ( pp(aa(fun(nat,nat),bool,order_mono(nat,nat),F11))
          & ( aa(set(nat),set(nat),image2(nat,nat,F11),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys2)) )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Ys2),aa(nat,nat,F11,I)) ) )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat))) )
              <=> ( aa(nat,nat,F11,I) = aa(nat,nat,F11,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_5856_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R2: set(C)] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G3),top_top(set(C)))),S))
       => finite673082921795544331dem_on(C,B,R2,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_5857_ring__1__class_Oof__int__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_nq(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_5858_image__mset_Oidentity,axiom,
    ! [A: $tType] : image_mset(A,A,aTP_Lamp_cc(A,A)) = id(multiset(A)) ).

% image_mset.identity
tff(fact_5859_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_5860_case__prod__Pair,axiom,
    ! [B: $tType,A: $tType] : 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)) = id(product_prod(A,B)) ).

% case_prod_Pair
tff(fact_5861_id__funpow,axiom,
    ! [A: $tType,N: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),id(A)) = id(A) ).

% id_funpow
tff(fact_5862_apfst__id,axiom,
    ! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).

% apfst_id
tff(fact_5863_apsnd__id,axiom,
    ! [B: $tType,A: $tType] : aa(fun(B,B),fun(product_prod(A,B),product_prod(A,B)),product_apsnd(B,B,A),id(B)) = id(product_prod(A,B)) ).

% apsnd_id
tff(fact_5864_comp__the__Some,axiom,
    ! [A: $tType] : aa(fun(A,option(A)),fun(A,A),comp(option(A),A,A,the2(A)),some(A)) = id(A) ).

% comp_the_Some
tff(fact_5865_mono__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(set(A),set(B)),A6: set(A),B5: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F3,A6)),aa(set(A),set(B),F3,B5)))) ) ).

% mono_Int
tff(fact_5866_rtranclp_Omono,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] : pp(aa(fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),bool,order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool))),aTP_Lamp_afp(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R3))) ).

% rtranclp.mono
tff(fact_5867_tranclp_Omono,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] : pp(aa(fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),bool,order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool))),aTP_Lamp_afq(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R3))) ).

% tranclp.mono
tff(fact_5868_mono__Un,axiom,
    ! [B: $tType,A: $tType,F3: fun(set(A),set(B)),A6: set(A),B5: set(A)] :
      ( pp(aa(fun(set(A),set(B)),bool,order_mono(set(A),set(B)),F3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F3,A6)),aa(set(A),set(B),F3,B5))),aa(set(A),set(B),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))) ) ).

% mono_Un
tff(fact_5869_sorted__remdups__adj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups_adj(A,Xs)) ) ) ).

% sorted_remdups_adj
tff(fact_5870_DEADID_Oin__rel,axiom,
    ! [B: $tType,A3: B,B2: B] :
      ( ( A3 = B2 )
    <=> ? [Z2: B] :
          ( pp(aa(set(B),bool,member(B,Z2),top_top(set(B))))
          & ( aa(B,B,id(B),Z2) = A3 )
          & ( aa(B,B,id(B),Z2) = B2 ) ) ) ).

% DEADID.in_rel
tff(fact_5871_option_Omap__id,axiom,
    ! [A: $tType,T5: option(A)] : aa(option(A),option(A),map_option(A,A,id(A)),T5) = T5 ).

% option.map_id
tff(fact_5872_option_Omap__id0,axiom,
    ! [A: $tType] : map_option(A,A,id(A)) = id(option(A)) ).

% option.map_id0
tff(fact_5873_map__fun_Oidentity,axiom,
    ! [B: $tType,A: $tType] : map_fun(A,A,B,B,aTP_Lamp_cc(A,A),aTP_Lamp_aej(B,B)) = id(fun(A,B)) ).

% map_fun.identity
tff(fact_5874_map__option_Oidentity,axiom,
    ! [A: $tType] : map_option(A,A,aTP_Lamp_cc(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_5875_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3) = id(A) ).

% funpow_simps_right(1)
tff(fact_5876_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_cc(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_5877_map__prod_Oidentity,axiom,
    ! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_cc(A,A),aTP_Lamp_aej(B,B)) = id(product_prod(A,B)) ).

% map_prod.identity
tff(fact_5878_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% remdups_adj_length
tff(fact_5879_set_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_cc(A,A)) = id(set(A)) ).

% set.identity
tff(fact_5880_apfst__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,C)] : product_apfst(A,C,B,F3) = product_map_prod(A,C,B,B,F3,id(B)) ).

% apfst_def
tff(fact_5881_apsnd__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] : aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3) = product_map_prod(A,A,B,C,id(A),F3) ).

% apsnd_def
tff(fact_5882_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ns(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_int_def
tff(fact_5883_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_nu(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_eq_int_def
tff(fact_5884_Id__on__Gr,axiom,
    ! [A: $tType,A6: set(A)] : id_on(A,A6) = bNF_Gr(A,A,A6,id(A)) ).

% Id_on_Gr
tff(fact_5885_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P2: fun(B,bool),Xs: list(B)] : foldr(B,A,F3,filter2(B,P2,Xs)) = foldr(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_afr(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F3),P2),Xs) ).

% foldr_filter
tff(fact_5886_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))) ).

% nat_def
tff(fact_5887_remdups__adj__Cons_H,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups_adj(A,dropWhile(A,aTP_Lamp_ap(A,fun(A,bool),X),Xs))) ).

% remdups_adj_Cons'
tff(fact_5888_fst__diag__id,axiom,
    ! [A: $tType,Z4: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_fst(A,A)),aTP_Lamp_ox(A,product_prod(A,A))),Z4) = aa(A,A,id(A),Z4) ).

% fst_diag_id
tff(fact_5889_snd__diag__id,axiom,
    ! [A: $tType,Z4: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_snd(A,A)),aTP_Lamp_ox(A,product_prod(A,A))),Z4) = aa(A,A,id(A),Z4) ).

% snd_diag_id
tff(fact_5890_remdups__adj__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),aTP_Lamp_afs(A,fun(A,fun(list(A),list(A))),X)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_5891_remdups__adj__adjacent,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I2) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I2)) ) ) ).

% remdups_adj_adjacent
tff(fact_5892_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : pp(aa(fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),bool,order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),aTP_Lamp_aft(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less))) ).

% ord.lexordp.mono
tff(fact_5893_finite_Omono,axiom,
    ! [A: $tType] : pp(aa(fun(fun(set(A),bool),fun(set(A),bool)),bool,order_mono(fun(set(A),bool),fun(set(A),bool)),aTP_Lamp_afu(fun(set(A),bool),fun(set(A),bool)))) ).

% finite.mono
tff(fact_5894_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys2: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_ap(A,fun(A,bool),Y),Ys2))) ).

% remdups_adj_append_dropWhile
tff(fact_5895_remdups__adj__append_H_H,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != nil(A) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,dropWhile(A,aTP_Lamp_afv(list(A),fun(A,bool),Xs),Ys2))) ) ) ).

% remdups_adj_append''
tff(fact_5896_tl__remdups__adj,axiom,
    ! [A: $tType,Ys2: list(A)] :
      ( ( Ys2 != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys2)) = remdups_adj(A,dropWhile(A,aTP_Lamp_afw(list(A),fun(A,bool),Ys2),aa(list(A),list(A),tl(A),Ys2))) ) ) ).

% tl_remdups_adj
tff(fact_5897_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => pp(aa(fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),bool,order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),aTP_Lamp_afx(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))))) ) ).

% lexordp.mono
tff(fact_5898_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))) ) ).

% remdups_adj_length_ge1
tff(fact_5899_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z4),A6) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_5900_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( finite_fold(A,B,F3,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z4,A6)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_5901_of__rat__def,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A) = aa(fun(product_prod(int,int),A),fun(rat,A),map_fun(rat,product_prod(int,int),A,A,rep_Rat,id(A)),aTP_Lamp_aeq(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_5902_positive__def,axiom,
    positive = aa(fun(product_prod(int,int),bool),fun(rat,bool),map_fun(rat,product_prod(int,int),bool,bool,rep_Rat,id(bool)),aTP_Lamp_acq(product_prod(int,int),bool)) ).

% positive_def
tff(fact_5903_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),bool),aTP_Lamp_afu(fun(set(A),bool),fun(set(A),bool))) ).

% finite_def
tff(fact_5904_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q2: fun(A,fun(B,C)),F3: fun(D,B)] :
          ( pp(aa(fun(A,fun(B,C)),bool,order_mono(A,fun(B,C)),Q2))
         => pp(aa(fun(A,fun(D,C)),bool,order_mono(A,fun(D,C)),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_afy(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q2),F3))) ) ) ).

% mono_compose
tff(fact_5905_less__integer__def,axiom,
    ord_less(code_integer) = aa(fun(int,fun(int,bool)),fun(code_integer,fun(code_integer,bool)),map_fun(code_integer,int,fun(int,bool),fun(code_integer,bool),code_int_of_integer,map_fun(code_integer,int,bool,bool,code_int_of_integer,id(bool))),ord_less(int)) ).

% less_integer_def
tff(fact_5906_less__eq__integer__def,axiom,
    ord_less_eq(code_integer) = aa(fun(int,fun(int,bool)),fun(code_integer,fun(code_integer,bool)),map_fun(code_integer,int,fun(int,bool),fun(code_integer,bool),code_int_of_integer,map_fun(code_integer,int,bool,bool,code_int_of_integer,id(bool))),ord_less_eq(int)) ).

% less_eq_integer_def
tff(fact_5907_plus__integer__def,axiom,
    plus_plus(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),plus_plus(int)) ).

% plus_integer_def
tff(fact_5908_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3)) = complete_lattice_lfp(A,F3) ) ) ) ).

% lfp_funpow
tff(fact_5909_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K2)),F3),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_5910_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: fun(A,bool)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ! [S6: A] :
                ( pp(aa(A,bool,P2,S6))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S6),complete_lattice_lfp(A,F3)))
                 => pp(aa(A,bool,P2,aa(A,A,F3,S6))) ) )
           => ( ! [M11: set(A)] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),M11))
                     => pp(aa(A,bool,P2,X5)) )
                 => pp(aa(A,bool,P2,aa(set(A),A,complete_Sup_Sup(A),M11))) )
             => pp(aa(A,bool,P2,complete_lattice_lfp(A,F3))) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_5911_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: A,F3: fun(A,A),P2: A] :
          ( ( A6 = complete_lattice_lfp(A,F3) )
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),P2))),P2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),P2)) ) ) ) ) ).

% def_lfp_induct
tff(fact_5912_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F3)),P2))),P2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),P2)) ) ) ) ).

% lfp_induct
tff(fact_5913_lfp__induct__set,axiom,
    ! [A: $tType,A3: A,F3: fun(set(A),set(A)),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,A3),complete_lattice_lfp(set(A),F3)))
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_lattice_lfp(set(A),F3)),aa(fun(A,bool),set(A),collect(A),P2)))))
             => pp(aa(A,bool,P2,X3)) )
         => pp(aa(A,bool,P2,A3)) ) ) ) ).

% lfp_induct_set
tff(fact_5914_def__lfp__induct__set,axiom,
    ! [A: $tType,A6: set(A),F3: fun(set(A),set(A)),A3: A,P2: fun(A,bool)] :
      ( ( A6 = complete_lattice_lfp(set(A),F3) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( pp(aa(set(A),bool,member(A,A3),A6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(fun(A,bool),set(A),collect(A),P2)))))
               => pp(aa(A,bool,P2,X3)) )
           => pp(aa(A,bool,P2,A3)) ) ) ) ) ).

% def_lfp_induct_set
tff(fact_5915_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X6: fun(bool,A),Y6: fun(bool,A)] :
          ( pp(aa(fun(bool,A),bool,aa(fun(bool,A),fun(fun(bool,A),bool),ord_less_eq(fun(bool,A)),X6),Y6))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fFalse)),aa(bool,A,Y6,fFalse)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fTrue)),aa(bool,A,Y6,fTrue))) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_5916_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,F3: fun(set(product_prod(A,B)),set(product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),complete_lattice_lfp(set(product_prod(A,B)),F3)))
     => ( pp(aa(fun(set(product_prod(A,B)),set(product_prod(A,B))),bool,order_mono(set(product_prod(A,B)),set(product_prod(A,B))),F3))
       => ( ! [A5: A,B4: B] :
              ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(set(product_prod(A,B)),set(product_prod(A,B)),F3,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))),complete_lattice_lfp(set(product_prod(A,B)),F3)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2))))))
             => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,A3),B2)) ) ) ) ).

% lfp_induct2
tff(fact_5917_lfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T5: A] : complete_lattice_lfp(A,aTP_Lamp_afz(A,fun(A,A),T5)) = T5 ) ).

% lfp_const
tff(fact_5918_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A6: A] :
          ( ! [U3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,U3)),U3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),U3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),complete_lattice_lfp(A,F3))) ) ) ).

% lfp_greatest
tff(fact_5919_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A6: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,A6)),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),A6)) ) ) ).

% lfp_lowerbound
tff(fact_5920_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),G3: fun(A,A)] :
          ( ! [Z8: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,Z8)),aa(A,A,G3,Z8)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),complete_lattice_lfp(A,G3))) ) ) ).

% lfp_mono
tff(fact_5921_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,fun(A,A))] :
          ( ! [X3: A,Y3: A,W: A,Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),W)),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_aga(fun(A,fun(A,A)),fun(A,A),F3)) = complete_lattice_lfp(A,aTP_Lamp_agb(fun(A,fun(A,A)),fun(A,A),F3)) ) ) ) ).

% lfp_lfp
tff(fact_5922_lfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G3: fun(A,B),F3: fun(B,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),G3))
         => ( pp(aa(fun(B,A),bool,order_mono(B,A),F3))
           => ( aa(A,B,G3,complete_lattice_lfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_agc(fun(A,B),fun(fun(B,A),fun(A,A)),G3),F3))) = complete_lattice_lfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_agd(fun(A,B),fun(fun(B,A),fun(B,B)),G3),F3)) ) ) ) ) ).

% lfp_rolling
tff(fact_5923_coinduct3__mono__lemma,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [F3: fun(A,set(B)),X6: set(B),B5: set(B)] :
          ( pp(aa(fun(A,set(B)),bool,order_mono(A,set(B)),F3))
         => pp(aa(fun(A,set(B)),bool,order_mono(A,set(B)),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_age(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),F3),X6),B5))) ) ) ).

% coinduct3_mono_lemma
tff(fact_5924_lfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),X: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F4))
         => ( ( aa(A,A,F4,X) = X )
           => ( ! [Z3: A] :
                  ( ( aa(A,A,F4,Z3) = Z3 )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z3)) )
             => ( complete_lattice_lfp(A,F4) = X ) ) ) ) ) ).

% lfp_eqI
tff(fact_5925_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] : complete_lattice_lfp(A,F3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_agf(fun(A,A),fun(A,bool),F3))) ) ).

% lfp_def
tff(fact_5926_plus__rat__def,axiom,
    plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_acl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_5927_inverse__rat__def,axiom,
    inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_acp(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_5928_times__rat__def,axiom,
    times_times(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_aco(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_5929_Fract__def,axiom,
    fract = aa(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,rat)),map_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),id(int),map_fun(int,int,product_prod(int,int),rat,id(int),abs_Rat)),aTP_Lamp_acm(int,fun(int,product_prod(int,int)))) ).

% Fract_def
tff(fact_5930_uminus__rat__def,axiom,
    uminus_uminus(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_acn(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_5931_Code__Numeral_Odup__def,axiom,
    code_dup = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),aTP_Lamp_abl(int,int)) ).

% Code_Numeral.dup_def
tff(fact_5932_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa2),Xa2))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa2)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,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(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_5933_positive_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( pp(aa(rat,bool,positive,aa(product_prod(int,int),rat,abs_Rat,X)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ).

% positive.abs_eq
tff(fact_5934_Fract_Orsp,axiom,
    pp(aa(fun(int,fun(int,product_prod(int,int))),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_acm(int,fun(int,product_prod(int,int)))),aTP_Lamp_acm(int,fun(int,product_prod(int,int))))) ).

% Fract.rsp
tff(fact_5935_dup_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,code_dup,X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,X)) ).

% dup.rep_eq
tff(fact_5936_dup_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,code_dup,aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),X)) ).

% dup.abs_eq
tff(fact_5937_of__rat_Orsp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(fun(product_prod(int,int),A),bool,aa(fun(product_prod(int,int),A),fun(fun(product_prod(int,int),A),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),A,A,ratrel,fequal(A)),aTP_Lamp_aeq(product_prod(int,int),A)),aTP_Lamp_aeq(product_prod(int,int),A))) ) ).

% of_rat.rsp
tff(fact_5938_ratrel__def,axiom,
    ! [X5: product_prod(int,int),Xa: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X5),Xa))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X5) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Xa) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X5)),aa(product_prod(int,int),int,product_snd(int,int),Xa)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X5)) ) ) ) ).

% ratrel_def
tff(fact_5939_times__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_aco(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_aco(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% times_rat.rsp
tff(fact_5940_uminus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_acn(product_prod(int,int),product_prod(int,int))),aTP_Lamp_acn(product_prod(int,int),product_prod(int,int)))) ).

% uminus_rat.rsp
tff(fact_5941_positive_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,ratrel,fequal(bool)),aTP_Lamp_acq(product_prod(int,int),bool)),aTP_Lamp_acq(product_prod(int,int),bool))) ).

% positive.rsp
tff(fact_5942_inverse__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_acp(product_prod(int,int),product_prod(int,int))),aTP_Lamp_acp(product_prod(int,int),product_prod(int,int)))) ).

% inverse_rat.rsp
tff(fact_5943_plus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_acl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_acl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% plus_rat.rsp
tff(fact_5944_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N))),one_one(code_integer)) ).

% Code_Numeral.sub_code(8)
tff(fact_5945_cr__rat__def,axiom,
    ! [X5: product_prod(int,int),Xa: rat] :
      ( cr_rat(X5,Xa)
    <=> ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X5),X5))
        & ( aa(product_prod(int,int),rat,abs_Rat,X5) = Xa ) ) ) ).

% cr_rat_def
tff(fact_5946_swap__comp__swap,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B)),comp(product_prod(B,A),product_prod(A,B),product_prod(A,B),product_swap(B,A)),product_swap(A,B)) = id(product_prod(A,B)) ).

% swap_comp_swap
tff(fact_5947_swap__swap,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P3)) = P3 ).

% swap_swap
tff(fact_5948_swap__simp,axiom,
    ! [A: $tType,B: $tType,X: B,Y: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X) ).

% swap_simp
tff(fact_5949_case__swap,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,fun(B,A)),P3: product_prod(C,B)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aTP_Lamp_agg(fun(C,fun(B,A)),fun(B,fun(C,A)),F3)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P3)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F3),P3) ).

% case_swap
tff(fact_5950_snd__swap,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),X)) = aa(product_prod(A,B),A,product_fst(A,B),X) ).

% snd_swap
tff(fact_5951_fst__swap,axiom,
    ! [A: $tType,B: $tType,X: product_prod(B,A)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),X)) = aa(product_prod(B,A),A,product_snd(B,A),X) ).

% fst_swap
tff(fact_5952_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Y: A,X: B,A6: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X)),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),A6)))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)),A6)) ) ).

% pair_in_swap_image
tff(fact_5953_surj__swap,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).

% surj_swap
tff(fact_5954_product__swap,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B5: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),product_Sigma(B,A,A6,aTP_Lamp_ws(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),A6)) ).

% product_swap
tff(fact_5955_prod_Oswap__def,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P3) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(product_prod(A,B),B,product_snd(A,B),P3)),aa(product_prod(A,B),A,product_fst(A,B),P3)) ).

% prod.swap_def
tff(fact_5956_Code__Numeral_Osub__def,axiom,
    code_sub = aa(fun(num,fun(num,int)),fun(num,fun(num,code_integer)),map_fun(num,num,fun(num,int),fun(num,code_integer),id(num),map_fun(num,num,int,code_integer,id(num),code_integer_of_int)),aTP_Lamp_abj(num,fun(num,int))) ).

% Code_Numeral.sub_def
tff(fact_5957_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_afx(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).

% ord_class.lexordp_def
tff(fact_5958_revg__fun,axiom,
    ! [A: $tType,A3: list(A),B2: list(A)] : revg(A,A3,B2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,A3)),B2) ).

% revg_fun
tff(fact_5959_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A),Y: A,Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
              & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2)) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_5960_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X3))
         => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ) ).

% lexordp_irreflexive
tff(fact_5961_revg_Osimps_I2_J,axiom,
    ! [A: $tType,A3: A,As: list(A),B2: list(A)] : revg(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As),B2) = revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),B2)) ).

% revg.simps(2)
tff(fact_5962_revg_Osimps_I1_J,axiom,
    ! [A: $tType,B2: list(A)] : revg(A,nil(A),B2) = B2 ).

% revg.simps(1)
tff(fact_5963_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_5964_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2))) ) ) ).

% lexordp.Cons
tff(fact_5965_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us: list(A),Vs: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs)))
         => ( ! [A5: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),A5))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),Vs)) ) ) ) ).

% lexordp_append_leftD
tff(fact_5966_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
         => ( ( ( A1 = nil(A) )
             => ! [Y3: A,Ys5: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
           => ( ! [X3: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
                 => ! [Y3: A] :
                      ( ? [Ys5: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3)) ) )
             => ~ ! [X3: A,Y3: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ! [Ys5: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                       => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X3))
                           => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys5)) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_5967_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs3),Ys4)) ) ) ) ) ).

% lexordp.simps
tff(fact_5968_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
         => ( ( ( Xs = nil(A) )
             => ! [Y3: A,Ys3: list(A)] : Ys2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
           => ( ! [X3: A] :
                  ( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4)
                 => ! [Y3: A] :
                      ( ? [Ys3: list(A)] : Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3)) ) )
             => ~ ! [X3: A,Xs4: list(A)] :
                    ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
                   => ! [Ys3: list(A)] :
                        ( ( Ys2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
                       => ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs4),Ys3)) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_5969_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A),P2: fun(list(A),fun(list(A),bool))] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
         => ( ! [Y3: A,Ys5: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P2,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))
           => ( ! [X3: A,Xs2: list(A),Y3: A,Ys5: list(A)] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5))) )
             => ( ! [X3: A,Xs2: list(A),Ys5: list(A)] :
                    ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys5))
                   => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P2,Xs2),Ys5))
                     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys5))) ) )
               => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P2,Xs),Ys2)) ) ) ) ) ) ).

% lexordp_induct
tff(fact_5970_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
        <=> ( ? [X4: A,Vs3: list(A)] : Ys2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Vs3))
            | ? [Us3: list(A),A9: A,B7: A,Vs3: list(A),Ws: list(A)] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A9),B7))
                & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A9),Vs3)) )
                & ( Ys2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),Ws)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_5971_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A,Us: list(A),Xs: list(A),Ys2: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)))) ) ) ).

% lexordp_append_left_rightI
tff(fact_5972_revg_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: list(A)] :
      ( ( revg(A,X,Xa2) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != Xa2 ) )
       => ~ ! [A5: A,As2: list(A)] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
             => ( Y != revg(A,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa2)) ) ) ) ) ).

% revg.elims
tff(fact_5973_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
        <=> pp(aa(set(product_prod(list(A),list(A))),bool,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),Ys2)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))))) ) ) ).

% lexordp_conv_lexord
tff(fact_5974_revg_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa2: list(A),Y: list(A)] :
      ( ( revg(A,X,Xa2) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_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)),X),Xa2)))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa2 )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_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)),Xa2))) ) )
         => ~ ! [A5: A,As2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
               => ( ( Y = revg(A,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa2)) )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),Xa2))) ) ) ) ) ) ).

% revg.pelims
tff(fact_5975_ord_Olexordp__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_aft(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less)) ).

% ord.lexordp_def
tff(fact_5976_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_agh(fun(A,set(B)),fun(A,filter(B)),F3)),X6)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X6))) ) ) ).

% INF_principal_finite
tff(fact_5977_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_mo(A,fun(A,bool))) = ord_max(A) ) ) ).

% dual_min
tff(fact_5978_subset__mset_Omin__arg__le_I2_J,axiom,
    ! [A: $tType,M: multiset(A),N: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N)))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N) = M ) ) ).

% subset_mset.min_arg_le(2)
tff(fact_5979_subset__mset_Omin__arg__le_I1_J,axiom,
    ! [A: $tType,N: multiset(A),M: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),N),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N)))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),N) = N ) ) ).

% subset_mset.min_arg_le(1)
tff(fact_5980_sup__principal,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),principal(A,A6)),principal(A,B5)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) ).

% sup_principal
tff(fact_5981_principal__le__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),principal(A,A6)),principal(A,B5)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ).

% principal_le_iff
tff(fact_5982_SUP__principal,axiom,
    ! [A: $tType,B: $tType,A6: fun(B,set(A)),I5: set(B)] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aTP_Lamp_agi(fun(B,set(A)),fun(B,filter(A)),A6)),I5)) = principal(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))) ).

% SUP_principal
tff(fact_5983_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = A3 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = B2 ) ) ) ).

% ord.min_def
tff(fact_5984_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_5985_subset__mset_Omin__add__distrib__left,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A),Z4: 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)),min(multiset(A),subseteq_mset(A)),X),Y)),Z4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Z4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y),Z4)) ).

% subset_mset.min_add_distrib_left
tff(fact_5986_subset__mset_Omin__add__distrib__right,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A),Z4: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),Y),Z4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Y)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Z4)) ).

% subset_mset.min_add_distrib_right
tff(fact_5987_at__bot__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_agj(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_5988_at__bot__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_agk(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C3))) ) ).

% at_bot_sub
tff(fact_5989_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_mo(A,fun(A,bool))) = ord_min(A) ) ) ).

% dual_max
tff(fact_5990_pick__same,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(code_natural,A,pick(A,aa(list(A),list(product_prod(code_natural,A)),map(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),one_one(code_natural))),Xs)),aa(nat,code_natural,code_natural_of_nat,L)) = aa(nat,A,nth(A,Xs),L) ) ) ).

% pick_same
tff(fact_5991_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = B2 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = A3 ) ) ) ).

% ord.max_def
tff(fact_5992_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_5993_less__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa2)),aa(nat,code_natural,code_natural_of_nat,X)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),X)) ) ).

% less_natural.abs_eq
tff(fact_5994_plus__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa2)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),X)) ).

% plus_natural.abs_eq
tff(fact_5995_less__eq__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa2)),aa(nat,code_natural,code_natural_of_nat,X)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X)) ) ).

% less_eq_natural.abs_eq
tff(fact_5996_subset__mset_Omax__add__distrib__left,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A),Z4: 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)),max(multiset(A),subseteq_mset(A)),X),Y)),Z4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),max(multiset(A),subseteq_mset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Z4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y),Z4)) ).

% subset_mset.max_add_distrib_left
tff(fact_5997_subset__mset_Omax__add__distrib__right,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A),Z4: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),max(multiset(A),subseteq_mset(A)),Y),Z4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),max(multiset(A),subseteq_mset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Y)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Z4)) ).

% subset_mset.max_add_distrib_right
tff(fact_5998_Predicate_Oiterate__upto_Opinduct,axiom,
    ! [A: $tType,A0: fun(code_natural,A),A1: code_natural,A22: code_natural,P2: fun(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)))] :
      ( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),A0),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),A1),A22))))
     => ( ! [F: fun(code_natural,A),N4: code_natural,M5: code_natural] :
            ( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N4),M5))))
           => ( ! [X5: product_unit] :
                  ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),M5),N4))
                 => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P2,F),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N4),one_one(code_natural))),M5)) )
             => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P2,F),N4),M5)) ) )
       => pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P2,A0),A1),A22)) ) ) ).

% Predicate.iterate_upto.pinduct
tff(fact_5999_size__multiset__add__mset,axiom,
    ! [A: $tType,F3: fun(A,nat),A3: A,M6: multiset(A)] : aa(multiset(A),nat,size_multiset(A,F3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M6)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,aa(A,nat,F3,A3))),aa(multiset(A),nat,size_multiset(A,F3),M6)) ).

% size_multiset_add_mset
tff(fact_6000_size__multiset__union,axiom,
    ! [A: $tType,F3: fun(A,nat),M6: multiset(A),N7: multiset(A)] : aa(multiset(A),nat,size_multiset(A,F3),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_multiset(A,F3),M6)),aa(multiset(A),nat,size_multiset(A,F3),N7)) ).

% size_multiset_union
tff(fact_6001_size__multiset__overloaded__def,axiom,
    ! [B: $tType] : size_size(multiset(B)) = size_multiset(B,aTP_Lamp_agl(B,nat)) ).

% size_multiset_overloaded_def
tff(fact_6002_image__o__collect,axiom,
    ! [B: $tType,C: $tType,A: $tType,G3: fun(C,B),F4: set(fun(A,set(C)))] : bNF_collect(A,B,aa(set(fun(A,set(C))),set(fun(A,set(B))),image2(fun(A,set(C)),fun(A,set(B)),comp(set(C),set(B),A,image2(C,B,G3))),F4)) = aa(fun(A,set(C)),fun(A,set(B)),comp(set(C),set(B),A,image2(C,B,G3)),bNF_collect(A,C,F4)) ).

% image_o_collect
tff(fact_6003_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_agm(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Sup_fin.eq_fold'
tff(fact_6004_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Sup_fin.singleton
tff(fact_6005_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)) = A3 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_6006_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_6007_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A6))) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_6008_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A6) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_6009_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_6010_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,member(A,A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)),X)) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_6011_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)),X))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A14),X)) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_6012_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A6) = aa(set(A),A,complete_Sup_Sup(A),A6) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_6013_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = aa(set(A),A,lattic5882676163264333800up_fin(A),X6) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_6014_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A6) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_6015_collect__def,axiom,
    ! [A: $tType,B: $tType,F4: set(fun(B,set(A))),X: B] : aa(B,set(A),bNF_collect(B,A,F4),X) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(B,set(A))),set(set(A)),image2(fun(B,set(A)),set(A),aTP_Lamp_agn(B,fun(fun(B,set(A)),set(A)),X)),F4)) ).

% collect_def
tff(fact_6016_collect__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: set(fun(C,set(B))),G3: fun(A,C)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,bNF_collect(C,B,F4)),G3) = bNF_collect(A,B,aa(set(fun(C,set(B))),set(fun(A,set(B))),image2(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_ago(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),G3)),F4)) ).

% collect_comp
tff(fact_6017_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5))) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_6018_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,finite_finite2(A),N7))
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic5882676163264333800up_fin(A),N7)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image2(A,A,H),N7)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_6019_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( B5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A6) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_6020_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_6021_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A6)),A6)) ) ) ) ) ).

% Sup_fin.closed
tff(fact_6022_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_6023_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = finite_fold(A,A,sup_sup(A),X,A6) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_6024_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_agp(set(A),fun(A,fun(A,bool)),A6),X))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_6025_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A6)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_agq(set(A),fun(set(A),fun(A,bool)),A6),B5))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_6026_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A6) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A6) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Sup_fin.remove
tff(fact_6027_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_6028_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_agr(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Inf_fin.eq_fold'
tff(fact_6029_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ags(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Max.eq_fold'
tff(fact_6030_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_lh(nat,fun(nat,bool),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_6031_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Max_singleton
tff(fact_6032_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Inf_fin.singleton
tff(fact_6033_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)),A3) = A3 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_6034_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,bool),set(int),collect(int),aTP_Lamp_lx(int,fun(int,bool),N))) = aa(int,int,abs_abs(int),N) ) ) ).

% Max_divisors_self_int
tff(fact_6035_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6036_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Max_less_iff
tff(fact_6037_Max__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A6: set(B),C3: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( A6 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_agt(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).

% Max_const
tff(fact_6038_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_6039_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6)) ) ) ) ) ).

% Max_insert
tff(fact_6040_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_6041_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A6)),A6)) ) ) ) ).

% Max_in
tff(fact_6042_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6)) = aa(set(A),A,lattic643756798349783984er_Max(A),A6) ) ) ) ) ).

% Max.in_idem
tff(fact_6043_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,lattic643756798349783984er_Max(A),A6))) ) ) ) ).

% Max.coboundedI
tff(fact_6044_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,finite_finite2(A),B5))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A6))
                 => ? [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),B5))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa)) ) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,member(A,X3),B5))
                   => ? [Xa: A] :
                        ( pp(aa(set(A),bool,member(A,Xa),A6))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa)) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = aa(set(A),A,lattic643756798349783984er_Max(A),B5) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6045_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [Y3: A] :
                ( pp(aa(set(A),bool,member(A,Y3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( pp(aa(set(A),bool,member(A,X),A6))
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_6046_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6))) ) ) ) ).

% Max_ge
tff(fact_6047_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)),A3)) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_6048_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A6) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_6049_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = M )
            <=> ( pp(aa(set(A),bool,member(A,M),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M)) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6050_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_6051_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A6) )
            <=> ( pp(aa(set(A),bool,member(A,M),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M)) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6052_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),X))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A14),X)) ) ) ) ) ) ).

% Max.boundedE
tff(fact_6053_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,member(A,A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),X)) ) ) ) ) ).

% Max.boundedI
tff(fact_6054_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_6055_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,member(A,B4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),A3)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6)) = A3 ) ) ) ) ).

% Max_insert2
tff(fact_6056_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = aa(set(A),A,lattic643756798349783984er_Max(A),X6) ) ) ) ) ).

% cSup_eq_Max
tff(fact_6057_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = aa(set(A),A,complete_Sup_Sup(A),A6) ) ) ) ) ).

% Max_Sup
tff(fact_6058_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_6059_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A14)) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_6060_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,member(A,A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6))) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_6061_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_6062_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X6) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_6063_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A6) = aa(set(A),A,complete_Inf_Inf(A),A6) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_6064_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A6) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_6065_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N) = aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_agu(int,fun(int,fun(int,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_6066_Sup__nat__def,axiom,
    ! [X6: set(nat)] :
      ( ( ( X6 = bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = zero_zero(nat) ) )
      & ( ( X6 != bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),X6) ) ) ) ).

% Sup_nat_def
tff(fact_6067_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N7: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M6),N7))
         => ( ( M6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N7))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M6)),aa(set(A),A,lattic643756798349783984er_Max(A),N7))) ) ) ) ) ).

% Max_mono
tff(fact_6068_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),aa(set(A),A,lattic643756798349783984er_Max(A),B5))) ) ) ) ) ).

% Max.subset_imp
tff(fact_6069_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,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),image2(A,A,H),N7)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_6070_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( B5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B5)),aa(set(A),A,lattic643756798349783984er_Max(A),A6)) = aa(set(A),A,lattic643756798349783984er_Max(A),A6) ) ) ) ) ) ).

% Max.subset
tff(fact_6071_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A6)),A6)) ) ) ) ) ).

% Max.closed
tff(fact_6072_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_6073_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798349783984er_Max(A),A6)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F3),A6)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_6074_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => ( ( B5 != 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)),A6),B5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),aa(set(A),A,lattic643756798349783984er_Max(A),B5)) ) ) ) ) ) ) ).

% Max.union
tff(fact_6075_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = finite_fold(A,A,ord_max(A),X,A6) ) ) ) ).

% Max.eq_fold
tff(fact_6076_card__le__Suc__Max,axiom,
    ! [S: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S)))) ) ).

% card_le_Suc_Max
tff(fact_6077_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6))) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_6078_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,finite_finite2(A),N7))
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic7752659483105999362nf_fin(A),N7)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image2(A,A,H),N7)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_6079_Max__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S: set(B),F3: fun(B,A),K2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( S != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_agv(fun(B,A),fun(A,fun(B,A)),F3),K2)),S)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image2(B,A,F3),S))),K2) ) ) ) ) ).

% Max_add_commute
tff(fact_6080_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( B5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A6) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_6081_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_6082_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)),A6)) ) ) ) ) ).

% Inf_fin.closed
tff(fact_6083_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_agw(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).

% divide_nat_def
tff(fact_6084_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_6085_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_agx(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6086_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = finite_fold(A,A,inf_inf(A),X,A6) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_6087_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)),aa(set(A),A,lattic5882676163264333800up_fin(A),A6))) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_6088_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_agy(set(A),fun(set(A),fun(A,bool)),A6),B5))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_6089_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A6)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_agz(set(A),fun(A,fun(A,bool)),A6),X))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_6090_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Max.insert_remove
tff(fact_6091_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Max.remove
tff(fact_6092_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_6093_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A6) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A6) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Inf_fin.remove
tff(fact_6094_sum__le__card__Max,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),groups7311177749621191930dd_sum(A,nat,F3,A6)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image2(A,nat,F3),A6))))) ) ).

% sum_le_card_Max
tff(fact_6095_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_mo(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_6096_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aha(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Min.eq_fold'
tff(fact_6097_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Min_singleton
tff(fact_6098_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6099_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_6100_Min__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A6: set(B),C3: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( A6 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_agt(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).

% Min_const
tff(fact_6101_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)) ) ) ) ) ).

% Min_insert
tff(fact_6102_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_6103_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_6104_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_6105_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),X)) ) ) ) ).

% Min_le
tff(fact_6106_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [Y3: A] :
                ( pp(aa(set(A),bool,member(A,Y3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( pp(aa(set(A),bool,member(A,X),A6))
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_6107_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),A3)) ) ) ) ).

% Min.coboundedI
tff(fact_6108_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A6)),A6)) ) ) ) ).

% Min_in
tff(fact_6109_linorder_OMin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).

% linorder.Min.cong
tff(fact_6110_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)) = aa(set(A),A,lattic643756798350308766er_Min(A),A6) ) ) ) ) ).

% Min.in_idem
tff(fact_6111_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,member(A,A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6))) ) ) ) ) ).

% Min.boundedI
tff(fact_6112_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)))
             => ! [A14: A] :
                  ( pp(aa(set(A),bool,member(A,A14),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A14)) ) ) ) ) ) ).

% Min.boundedE
tff(fact_6113_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A6) )
            <=> ( pp(aa(set(A),bool,member(A,M),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X4)) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6114_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Min_le_iff
tff(fact_6115_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = M )
            <=> ( pp(aa(set(A),bool,member(A,M),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X4)) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6116_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Min_less_iff
tff(fact_6117_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,member(A,B4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B4)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6)) = A3 ) ) ) ) ).

% Min_insert2
tff(fact_6118_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = aa(set(A),A,complete_Inf_Inf(A),A6) ) ) ) ) ).

% Min_Inf
tff(fact_6119_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = aa(set(A),A,lattic643756798350308766er_Min(A),X6) ) ) ) ) ).

% cInf_eq_Min
tff(fact_6120_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_6121_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A6))) ) ) ) ) ).

% Min.subset_imp
tff(fact_6122_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N7: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M6),N7))
         => ( ( M6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N7))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N7)),aa(set(A),A,lattic643756798350308766er_Min(A),M6))) ) ) ) ) ).

% Min_antimono
tff(fact_6123_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N7: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( pp(aa(set(A),bool,finite_finite2(A),N7))
           => ( ( N7 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798350308766er_Min(A),N7)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,H),N7)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_6124_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( B5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A6)) = aa(set(A),A,lattic643756798350308766er_Min(A),A6) ) ) ) ) ) ).

% Min.subset
tff(fact_6125_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A))))))
             => pp(aa(set(A),bool,member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A6)),A6)) ) ) ) ) ).

% Min.closed
tff(fact_6126_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_6127_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798350308766er_Min(A),A6)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F3),A6)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_6128_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B5))
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),aa(set(A),A,lattic643756798350308766er_Min(A),B5)) ) ) ) ) ) ) ).

% Min.union
tff(fact_6129_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = finite_fold(A,A,ord_min(A),X,A6) ) ) ) ).

% Min.eq_fold
tff(fact_6130_Min__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S: set(B),F3: fun(B,A),K2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ( S != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_agv(fun(B,A),fun(A,fun(B,A)),F3),K2)),S)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image2(B,A,F3),S))),K2) ) ) ) ) ).

% Min_add_commute
tff(fact_6131_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = X ) )
            & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Min.insert_remove
tff(fact_6132_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = X ) )
              & ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Min.remove
tff(fact_6133_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
                & pp(aa(A,bool,P2,X5)) )
           => ( find(A,P2,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ahb(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P2)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_6134_card__Min__le__sum,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image2(A,nat,F3),A6)))),groups7311177749621191930dd_sum(A,nat,F3,A6))) ) ).

% card_Min_le_sum
tff(fact_6135_Gcd__eq__Max,axiom,
    ! [M6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M6))
     => ( ( M6 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M6))
         => ( gcd_Gcd(nat,M6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),aTP_Lamp_ahc(nat,set(nat))),M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_6136_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_mo(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_6137_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( ( gcd_Gcd(A,A6) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_6138_linorder_OMax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).

% linorder.Max.cong
tff(fact_6139_Gcd__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Gcd(A,aa(set(B),set(A),image2(B,A,F3),A6))),gcd_Gcd(A,aa(set(B),set(A),image2(B,A,G3),A6)))) ) ) ).

% Gcd_mono
tff(fact_6140_INT__greaterThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).

% INT_greaterThan_UNIV
tff(fact_6141_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ahd(nat,bool))) ) ).

% semiring_char_def
tff(fact_6142_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_greaterThan(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),I2)) ) ) ).

% greaterThan_iff
tff(fact_6143_cInf__greaterThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A)
        & no_top(A) )
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_greaterThan(A),X)) = X ) ).

% cInf_greaterThan
tff(fact_6144_Gcd__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,aa(set(int),set(int),image2(int,int,abs_abs(int)),K5)) = gcd_Gcd(int,K5) ).

% Gcd_abs_eq
tff(fact_6145_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),X)),aa(A,set(A),set_ord_greaterThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% greaterThan_subset_iff
tff(fact_6146_Gcd__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_ahe(int,nat)),K5)) = aa(int,nat,nat2,gcd_Gcd(int,K5)) ).

% Gcd_nat_abs_eq
tff(fact_6147_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),top_top(A)))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_6148_Gcd__int__eq,axiom,
    ! [N7: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N7)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N7)) ).

% Gcd_int_eq
tff(fact_6149_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_6150_Gcd__int__greater__eq__0,axiom,
    ! [K5: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K5))) ).

% Gcd_int_greater_eq_0
tff(fact_6151_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_6152_Gcd__int__def,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),K5))) ).

% Gcd_int_def
tff(fact_6153_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A5: A,B4: A,X3: A] :
              ( pp(aa(set(A),bool,member(A,A5),S))
             => ( pp(aa(set(A),bool,member(A,B4),S))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B4))
                   => pp(aa(set(A),bool,member(A,X3),S)) ) ) ) )
         => ? [A5: A,B4: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = aa(A,set(A),set_ord_lessThan(A),B4) )
              | ( S = aa(A,set(A),set_ord_atMost(A),B4) )
              | ( S = aa(A,set(A),set_ord_greaterThan(A),A5) )
              | ( S = aa(A,set(A),set_ord_atLeast(A),A5) )
              | ( S = set_or5935395276787703475ssThan(A,A5,B4) )
              | ( S = set_or3652927894154168847AtMost(A,A5,B4) )
              | ( S = set_or7035219750837199246ssThan(A,A5,B4) )
              | ( S = set_or1337092689740270186AtMost(A,A5,B4) ) ) ) ) ).

% interval_cases
tff(fact_6154_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),T5: list(product_prod(A,C)),K2: A,X: C] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,T5),K2) = aa(C,option(C),some(C),X) )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_ahf(fun(A,B),fun(A,fun(C,product_prod(B,C))),F3))),T5)),aa(A,B,F3,K2)) = aa(C,option(C),some(C),X) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_6155_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,member(A,I2),aa(A,set(A),set_ord_atLeast(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),I2)) ) ) ).

% atLeast_iff
tff(fact_6156_cInf__atLeast,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atLeast(A),X)) = X ) ).

% cInf_atLeast
tff(fact_6157_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),X)),aa(A,set(A),set_ord_atLeast(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% atLeast_subset_iff
tff(fact_6158_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K2)),aa(A,set(A),set_ord_atLeast(A),I2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),I2)) ) ).

% image_add_atLeast
tff(fact_6159_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L4: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atLeast(A),L4)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L4),L)) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_6160_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_ahg(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6161_inj__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),top_top(set(product_prod(A,B))))
    <=> inj_on(A,C,F3,top_top(set(A))) ) ).

% inj_apfst
tff(fact_6162_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F3,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_6163_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C),A6: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),product_Sigma(A,B,A6,aTP_Lamp_abb(A,set(B))))
    <=> inj_on(A,C,F3,A6) ) ).

% inj_on_apfst
tff(fact_6164_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C),A6: set(B)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aaz(set(B),fun(A,set(B)),A6)))
    <=> inj_on(B,C,F3,A6) ) ).

% inj_on_apsnd
tff(fact_6165_inj__on__Un__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))
     => ( ( aa(set(A),set(B),image2(A,B,F3),A6) = aa(set(A),set(B),image2(A,B,F3),B5) )
      <=> ( A6 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_6166_prod_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),A6: set(B)] :
          ( inj_on(B,A,G3,A6)
         => ( groups7121269368397514597t_prod(A,A,aTP_Lamp_ahh(A,A),aa(set(B),set(A),image2(B,A,G3),A6)) = groups7121269368397514597t_prod(B,A,G3,A6) ) ) ) ).

% prod.image_eq
tff(fact_6167_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(B,A),T8: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(B),set(A),image2(B,A,F3),T8)))
    <=> ? [U6: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U6),T8))
          & inj_on(B,A,F3,U6)
          & ( S = aa(set(B),set(A),image2(B,A,F3),U6) ) ) ) ).

% subset_image_inj
tff(fact_6168_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A6: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4))
         => ( ( aa(set(A),set(B),image2(A,B,F3),A6) = aa(set(A),set(B),image2(A,B,F3),B5) )
          <=> ( A6 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_6169_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(A),A3: A,A6: set(A)] :
      ( inj_on(A,B,F3,B5)
     => ( pp(aa(set(A),bool,member(A,A3),B5))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => ( pp(aa(set(B),bool,member(B,aa(A,B,F3,A3)),aa(set(A),set(B),image2(A,B,F3),A6)))
          <=> pp(aa(set(A),bool,member(A,A3),A6)) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_6170_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(A),A6: set(A)] :
      ( inj_on(A,B,F3,B5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),aa(set(A),set(B),image2(A,B,F3),B5))) ) ) ).

% inj_on_strict_subset
tff(fact_6171_sum_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A6: set(B)] :
          ( inj_on(B,A,G3,A6)
         => ( groups7311177749621191930dd_sum(A,A,aTP_Lamp_tn(A,A),aa(set(B),set(A),image2(B,A,G3),A6)) = groups7311177749621191930dd_sum(B,A,G3,A6) ) ) ) ).

% sum.image_eq
tff(fact_6172_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L4: A,H5: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),set_or1337092689740270186AtMost(A,L4,H5))) ) ).

% not_Ici_le_Icc
tff(fact_6173_inj__on__diff__nat,axiom,
    ! [N7: set(nat),K2: nat] :
      ( ! [N4: nat] :
          ( pp(aa(set(nat),bool,member(nat,N4),N7))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N4)) )
     => inj_on(nat,nat,aTP_Lamp_wj(nat,fun(nat,nat),K2),N7) ) ).

% inj_on_diff_nat
tff(fact_6174_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( inj_on(B,A,F3,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ahi(set(A),fun(fun(B,A),fun(B,bool)),S),F3)))) ) ) ).

% finite_Collect
tff(fact_6175_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( inj_on(B,A,F3,top_top(set(B)))
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ahi(set(A),fun(fun(B,A),fun(B,bool)),A6),F3)))) ) ) ).

% finite_inverse_image
tff(fact_6176_map__prod__inj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(A,B),A6: set(A),G3: fun(C,D),B5: set(C)] :
      ( inj_on(A,B,F3,A6)
     => ( inj_on(C,D,G3,B5)
       => inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F3,G3),product_Sigma(A,C,A6,aTP_Lamp_abr(set(C),fun(A,set(C)),B5))) ) ) ).

% map_prod_inj_on
tff(fact_6177_inj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( inj_on(A,A,F3,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(set(A))) ) ).

% inj_fn
tff(fact_6178_inj__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => inj_on(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ).

% inj_of_nat
tff(fact_6179_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
             => ( aa(A,B,F3,X3) != aa(A,B,F3,Y3) ) )
         => inj_on(A,B,F3,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_6180_inj__Some,axiom,
    ! [A: $tType,A6: set(A)] : inj_on(A,option(A),some(A),A6) ).

% inj_Some
tff(fact_6181_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A6: set(A)] : inj_on(A,A,aTP_Lamp_ahj(A,fun(A,A),A3),A6) ) ).

% inj_on_add'
tff(fact_6182_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A6: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),A6) ) ).

% inj_on_add
tff(fact_6183_inj__on__id2,axiom,
    ! [A: $tType,A6: set(A)] : inj_on(A,A,aTP_Lamp_cc(A,A),A6) ).

% inj_on_id2
tff(fact_6184_inj__Suc,axiom,
    ! [N7: set(nat)] : inj_on(nat,nat,suc,N7) ).

% inj_Suc
tff(fact_6185_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),top_top(set(A))) ) ).

% inj_add_left
tff(fact_6186_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(B,A),D5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( inj_on(B,A,F3,D5)
       => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_ahk(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),A6),F3),D5)))) ) ) ).

% finite_inverse_image_gen
tff(fact_6187_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_pk(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_6188_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(option(A),option(B),map_option(A,B,F3),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_6189_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_ahl(fun(A,B),fun(A,fun(C,B)),F3),top_top(set(A))) ) ).

% inj_fun
tff(fact_6190_inj__singleton,axiom,
    ! [A: $tType,A6: set(A)] : inj_on(A,set(A),aTP_Lamp_wt(A,set(A)),A6) ).

% inj_singleton
tff(fact_6191_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_vo(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_6192_inj__on__fst__map__to__set,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),map_to_set(A,B,M)) ).

% inj_on_fst_map_to_set
tff(fact_6193_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A6: set(A),F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
             => ( pp(aa(set(A),bool,member(A,X3),A6))
               => ( pp(aa(set(A),bool,member(A,Y3),A6))
                 => ( aa(A,B,F3,X3) != aa(A,B,F3,Y3) ) ) ) )
         => ( ! [X3: A,Y3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A6))
               => ( pp(aa(set(A),bool,member(A,Y3),A6))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3)) ) ) )
           => inj_on(A,B,F3,A6) ) ) ) ).

% linorder_inj_onI
tff(fact_6194_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_6195_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(A),A6: set(A)] :
      ( inj_on(A,B,F3,B5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
       => inj_on(A,B,F3,A6) ) ) ).

% subset_inj_on
tff(fact_6196_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
       => inj_on(A,B,F3,B5) ) ) ).

% inj_on_subset
tff(fact_6197_inj__swap,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_swap(A,B),A6) ).

% inj_swap
tff(fact_6198_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_we(fun(A,B),fun(A,product_prod(A,B)),F3),X6) ).

% inj_on_convol_ident
tff(fact_6199_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C3: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_we(fun(A,B),fun(A,product_prod(A,B)),C3),S) ).

% inj_Pair(1)
tff(fact_6200_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C3: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_ahm(fun(A,B),fun(A,product_prod(B,A)),C3),S) ).

% inj_Pair(2)
tff(fact_6201_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atLeast(A),L))) ) ).

% not_UNIV_le_Ici
tff(fact_6202_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_pc(A,fun(B,product_prod(B,A)))),A6) ).

% swap_inj_on
tff(fact_6203_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H5: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),H5))) ) ).

% not_Ici_le_Iic
tff(fact_6204_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L4: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),aa(A,set(A),set_ord_atLeast(A),L4))) ) ).

% not_Iic_le_Ici
tff(fact_6205_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_atLeast(A),A3))) ) ).

% Ioi_le_Ico
tff(fact_6206_inj__split__Cons,axiom,
    ! [A: $tType,X6: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_so(list(A),fun(A,list(A)))),X6) ).

% inj_split_Cons
tff(fact_6207_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A6: set(A),A13: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => ( ? [F11: fun(A,B)] :
            ( inj_on(A,B,F11,A6)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F11),A6)),A13)) )
      <=> ? [G7: fun(B,A)] : aa(set(B),set(A),image2(B,A,G7),A13) = A6 ) ) ).

% inj_on_iff_surj
tff(fact_6208_endo__inj__surj,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image2(A,A,F3),A6)),A6))
       => ( inj_on(A,A,F3,A6)
         => ( aa(set(A),set(A),image2(A,A,F3),A6) = A6 ) ) ) ) ).

% endo_inj_surj
tff(fact_6209_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),B5))
       => ( pp(aa(set(B),bool,finite_finite2(B),B5))
         => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ) ).

% inj_on_finite
tff(fact_6210_finite__surj__inj,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),image2(A,A,F3),A6)))
       => inj_on(A,A,F3,A6) ) ) ).

% finite_surj_inj
tff(fact_6211_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),aa(set(A),set(B),image2(A,B,F3),B5)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ) ).

% inj_image_subset_iff
tff(fact_6212_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A6: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4))
         => ( aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),aa(set(A),set(B),image2(A,B,F3),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_6213_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A6: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)),C4))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C4))
         => ( aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),aa(set(A),set(B),image2(A,B,F3),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_6214_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: set(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(B),nat,finite_card(B),A6)))
     => ~ inj_on(B,A,F3,A6) ) ).

% pigeonhole
tff(fact_6215_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys2: list(A)] :
      ( inj_on(A,B,F3,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),Ys2)))
     => ( ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys2) )
      <=> ( Xs = Ys2 ) ) ) ).

% inj_on_map_eq_map
tff(fact_6216_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys2) )
     => ( inj_on(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys2)))
       => ( Xs = Ys2 ) ) ) ).

% map_inj_on
tff(fact_6217_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_aho(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_6218_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),F3: fun(B,C)] :
      ( ! [I3: A,J3: A] :
          ( pp(aa(set(A),bool,member(A,I3),I5))
         => ( pp(aa(set(A),bool,member(A,J3),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,I3)),aa(A,set(B),A6,J3)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,J3)),aa(A,set(B),A6,I3))) ) ) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => inj_on(B,C,F3,aa(A,set(B),A6,I3)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_6219_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( filter2(A,aa(A,fun(A,bool),aTP_Lamp_ahp(fun(A,B),fun(A,fun(A,bool)),F3),Y),Xs) = filter2(A,aa(A,fun(A,bool),fequal(A),Y),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_6220_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F3: fun(B,C),A6: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => inj_on(B,C,F3,aa(A,set(B),A6,I3)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5))) ) ) ).

% inj_on_INTER
tff(fact_6221_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ? [N4: nat,F: fun(nat,A)] :
          ( ( A6 = aa(set(nat),set(A),image2(nat,A,F),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N4))) )
          & inj_on(nat,A,F,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N4))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6222_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ~ ! [F: fun(A,nat)] :
            ( ? [N4: nat] : aa(set(A),set(nat),image2(A,nat,F),A6) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N4))
           => ~ inj_on(A,nat,F,A6) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_6223_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ? [F: fun(A,nat),N4: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F),A6) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N4)) )
          & inj_on(A,nat,F,A6) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6224_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_6225_set__to__map__inverse,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( inj_on(product_prod(A,B),A,product_fst(A,B),S)
     => ( map_to_set(A,B,set_to_map(A,B,S)) = S ) ) ).

% set_to_map_inverse
tff(fact_6226_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),A3)),aa(A,set(A),set_ord_greaterThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6227_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),B5: set(B),G3: fun(B,A)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),B5))
       => ( inj_on(B,A,G3,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G3),B5)),A6))
           => ( pp(aa(set(A),bool,finite_finite2(A),A6))
             => ( pp(aa(set(B),bool,finite_finite2(B),B5))
               => ( aa(set(A),nat,finite_card(A),A6) = aa(set(B),nat,finite_card(B),B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_6228_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S: set(A),T8: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(B),bool,finite_finite2(B),T8))
       => ( ( aa(set(A),nat,finite_card(A),S) = aa(set(B),nat,finite_card(B),T8) )
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),S)),T8))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),T8))
                 => ? [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),S))
                      & ( aa(A,B,F3,Xa3) = X4 ) ) )
            <=> inj_on(A,B,F3,S) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_6229_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(B),A6: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F3),A6)))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)),A6)) ) ) ).

% vimage_subsetI
tff(fact_6230_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),uminus_uminus(set(A)),A6))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)))) ) ).

% inj_image_Compl_subset
tff(fact_6231_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X3: nat] :
            ( pp(aa(set(nat),bool,member(nat,X3),I5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs))) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_6232_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
    <=> ? [F11: fun(nat,A)] :
          ( inj_on(nat,A,F11,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F11),top_top(set(nat)))),S)) ) ) ).

% infinite_iff_countable_subset
tff(fact_6233_infinite__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
     => ? [F: fun(nat,A)] :
          ( inj_on(nat,A,F,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F),top_top(set(nat)))),S)) ) ) ).

% infinite_countable_subset
tff(fact_6234_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),G3: fun(A,B),B5: set(A)] :
      ( inj_on(A,B,F3,A6)
     => ( inj_on(A,B,G3,B5)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),aa(set(A),set(B),image2(A,B,G3),B5)) = 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_ahq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F3),A6),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_6235_set__to__map__simp,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B)),K2: A,V2: B] :
      ( inj_on(product_prod(A,B),A,product_fst(A,B),S)
     => ( ( aa(A,option(B),set_to_map(A,B,S),K2) = aa(B,option(B),some(B),V2) )
      <=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V2)),S)) ) ) ).

% set_to_map_simp
tff(fact_6236_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),C4: set(A),A6: set(C),B5: fun(C,set(A)),J2: C] :
      ( inj_on(A,B,F3,C4)
     => ( ! [X3: C] :
            ( pp(aa(set(C),bool,member(C,X3),A6))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(C,set(A),B5,X3)),C4)) )
       => ( pp(aa(set(C),bool,member(C,J2),A6))
         => ( aa(set(A),set(B),image2(A,B,F3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B5),A6))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_ahr(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F3),B5)),A6)) ) ) ) ) ).

% image_INT
tff(fact_6237_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A),S2: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),S2) = S2 )
     => ( ! [M5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M5))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M5),F3),S2) != S2 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_ahs(fun(A,A),fun(A,fun(nat,A)),F3),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_6238_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => ( ? [F11: fun(A,B)] :
              ( inj_on(A,B,F11,A6)
              & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F11),A6)),B5)) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B5))) ) ) ) ).

% inj_on_iff_card_le
tff(fact_6239_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),B5))
       => ( pp(aa(set(B),bool,finite_finite2(B),B5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B5))) ) ) ) ).

% card_inj_on_le
tff(fact_6240_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B5)))
         => ? [F: fun(A,B)] :
              ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),A6)),B5))
              & inj_on(A,B,F,A6) ) ) ) ) ).

% card_le_inj
tff(fact_6241_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))
    <=> ( inj_on(A,B,F3,A6)
        & inj_on(A,B,F3,B5)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5))),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A6))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_6242_ivl__disj__un__singleton_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_6243_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_6244_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),aa(set(A),set(B),image2(A,B,F3),top_top(set(A)))))
       => ( aa(set(A),nat,finite_card(A),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)) = aa(set(B),nat,finite_card(B),A6) ) ) ) ).

% card_vimage_inj
tff(fact_6245_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),D5: set(A),A6: set(B)] :
      ( inj_on(A,B,F3,D5)
     => ( pp(aa(set(B),bool,finite_finite2(B),A6))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),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(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),A6)),D5))),aa(set(B),nat,finite_card(B),A6))) ) ) ).

% card_vimage_inj_on_le
tff(fact_6246_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_6247_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A6: fun(B,set(A)),I5: set(B)] :
    ? [F: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5)))
      & pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),aa(set(A),set(product_prod(B,A)),image2(A,product_prod(B,A),F),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5)))),product_Sigma(B,A,I5,A6))) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_6248_UN__atLeast__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atLeast_UNIV
tff(fact_6249_map__sorted__distinct__set__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Ys2: list(B)] :
          ( inj_on(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys2)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
           => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
             => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Ys2))
               => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Ys2))
                 => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys2) )
                   => ( Xs = Ys2 ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_6250_sort__key__inj__key__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),Ys2: list(B),F3: fun(B,A)] :
          ( ( mset(B,Xs) = mset(B,Ys2) )
         => ( inj_on(B,A,F3,aa(list(B),set(B),set2(B),Xs))
           => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Ys2))
             => ( aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs) = Ys2 ) ) ) ) ) ).

% sort_key_inj_key_eq
tff(fact_6251_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F3: fun(A,B),G3: fun(C,B),A6: set(A),B5: set(C)] :
          ( inj_on(product_prod(A,C),B,aa(fun(A,fun(C,B)),fun(product_prod(A,C),B),product_case_prod(A,C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ga(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F3),G3)),product_Sigma(A,C,A6,aTP_Lamp_abr(set(C),fun(A,set(C)),B5)))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),groups7311177749621191930dd_sum(A,B,F3,A6)),groups7311177749621191930dd_sum(C,B,G3,B5)) = groups7311177749621191930dd_sum(B,B,id(B),aa(fun(B,bool),set(B),collect(B),aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_aht(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F3),G3),A6),B5))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_6252_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),X: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_ahu(fun(A,A),fun(A,fun(A,bool)),P3),X))))
       => ~ ! [N4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N4),P3),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_6253_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: B] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the(A,aa(B,fun(A,bool),aTP_Lamp_ahv(fun(A,B),fun(B,fun(A,bool)),F3),A3))),bot_bot(set(A))))) ) ).

% inj_vimage_singleton
tff(fact_6254_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),A3: B] :
      ( inj_on(A,B,F3,A6)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))),A6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ahw(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F3),A6),A3))),bot_bot(set(A))))) ) ).

% inj_on_vimage_singleton
tff(fact_6255_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A6))
     => ( inj_on(A,B,F3,A6)
       => ( aa(list(B),list(A),map(B,A,inv_on(A,B,F3,A6)),aa(list(A),list(B),map(A,B,F3),L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_6256_at__top__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_ahx(A,filter(A))),top_top(set(A)))) ) ) ).

% at_top_def
tff(fact_6257_inv__on__f__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),X: A] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(A),bool,member(A,X),A6))
       => ( aa(B,A,inv_on(A,B,F3,A6),aa(A,B,F3,X)) = X ) ) ) ).

% inv_on_f_f
tff(fact_6258_f__inv__on__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),image2(B,A,F3),A6)))
     => ( aa(B,A,F3,aa(A,B,inv_on(B,A,F3,A6),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_6259_inv__on__f__range,axiom,
    ! [A: $tType,B: $tType,Y: A,F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),image2(B,A,F3),A6)))
     => pp(aa(set(B),bool,member(B,aa(A,B,inv_on(B,A,F3,A6),Y)),A6)) ) ).

% inv_on_f_range
tff(fact_6260_at__top__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_ahy(A,filter(A))),aa(A,set(A),set_ord_atLeast(A),C3))) ) ).

% at_top_sub
tff(fact_6261_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),X: B,Y: A,Z4: A] :
      ( ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y) )
     => ( inj_on(B,option(A),M,dom(B,A,M))
       => ( ~ pp(aa(set(A),bool,member(A,Z4),ran(B,A,M)))
         => ( ran(B,A,fun_upd(B,option(A),M,X,aa(A,option(A),some(A),Z4))) = 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)),minus_minus(set(A)),ran(B,A,M)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z4),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_6262_If__the__inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,I2: A,C4: set(A),G3: fun(A,B),X: A] :
      ( pp(aa(set(A),bool,member(A,I2),C4))
     => ( inj_on(A,B,G3,C4)
       => ( aa(A,A,aa(fun(A,B),fun(A,A),comp(B,A,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_ahz(set(A),fun(fun(A,B),fun(A,fun(B,A))),C4),G3),X)),G3),I2) = aa(A,A,id(A),I2) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_6263_restrict__map__self,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : restrict_map(A,B,M,dom(A,B,M)) = M ).

% restrict_map_self
tff(fact_6264_dom__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),N: fun(A,option(B))] : dom(A,B,map_add(A,B,M,N)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,N)),dom(A,B,M)) ).

% dom_map_add
tff(fact_6265_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_bp(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_6266_map__update__eta__repair_I1_J,axiom,
    ! [B: $tType,A: $tType,K2: A,V2: B,M: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_aia(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),K2),V2),M)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K2),dom(A,B,M)) ).

% map_update_eta_repair(1)
tff(fact_6267_dom__const_H,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : dom(A,B,aTP_Lamp_aib(fun(A,B),fun(A,option(B)),F3)) = top_top(set(A)) ).

% dom_const'
tff(fact_6268_restrict__map__inv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X5: A] : aa(A,option(B),restrict_map(A,B,F3,aa(set(A),set(A),uminus_uminus(set(A)),dom(A,B,F3))),X5) = none(B) ).

% restrict_map_inv
tff(fact_6269_map__add__upd__left,axiom,
    ! [A: $tType,B: $tType,M: A,E23: fun(A,option(B)),E12: fun(A,option(B)),U1: B] :
      ( ~ pp(aa(set(A),bool,member(A,M),dom(A,B,E23)))
     => ( map_add(A,B,fun_upd(A,option(B),E12,M,aa(B,option(B),some(B),U1)),E23) = fun_upd(A,option(B),map_add(A,B,E12,E23),M,aa(B,option(B),some(B),U1)) ) ) ).

% map_add_upd_left
tff(fact_6270_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xs: list(A),Ys2: list(B)] : dom(A,B,map_upds(A,B,M,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))),dom(A,B,M)) ).

% dom_map_upds
tff(fact_6271_ran__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,M1,M22)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,M1)),ran(A,B,M22)) ) ) ).

% ran_map_add
tff(fact_6272_ran__add,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F3)),dom(A,B,G3)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,F3,G3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F3)),ran(A,B,G3)) ) ) ).

% ran_add
tff(fact_6273_insert__dom,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),X: B,Y: A] :
      ( ( aa(B,option(A),F3,X) = aa(A,option(A),some(A),Y) )
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),dom(B,A,F3)) = dom(B,A,F3) ) ) ).

% insert_dom
tff(fact_6274_dom__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_su(fun(A,option(B)),fun(A,bool),M)) ).

% dom_def
tff(fact_6275_domI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M,A3) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(B),bool,member(B,A3),dom(B,A,M))) ) ).

% domI
tff(fact_6276_domD,axiom,
    ! [A: $tType,B: $tType,A3: A,M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,member(A,A3),dom(A,B,M)))
     => ? [B4: B] : aa(A,option(B),M,A3) = aa(B,option(B),some(B),B4) ) ).

% domD
tff(fact_6277_nempty__dom,axiom,
    ! [B: $tType,A: $tType,E3: fun(A,option(B))] :
      ( ~ ! [X5: A] : aa(A,option(B),E3,X5) = none(B)
     => ~ ! [M5: A] : ~ pp(aa(set(A),bool,member(A,M5),dom(A,B,E3))) ) ).

% nempty_dom
tff(fact_6278_dom__map__option,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,fun(C,B)),M: fun(A,option(C))] : dom(A,B,aa(fun(A,option(C)),fun(A,option(B)),aTP_Lamp_aic(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),F3),M)) = dom(A,C,M) ).

% dom_map_option
tff(fact_6279_map__dom__ran__finite,axiom,
    ! [B: $tType,A: $tType,M6: fun(A,option(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M6)))
     => pp(aa(set(B),bool,finite_finite2(B),ran(A,B,M6))) ) ).

% map_dom_ran_finite
tff(fact_6280_le__map__dom__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [M: fun(A,option(B)),M8: fun(A,option(B))] :
          ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M8))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),dom(A,B,M)),dom(A,B,M8))) ) ) ).

% le_map_dom_mono
tff(fact_6281_dom__if,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),F3: fun(A,option(B)),G3: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_aid(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),P2),F3),G3)) = 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)),dom(A,B,F3)),aa(fun(A,bool),set(A),collect(A),P2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,G3)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2)))) ).

% dom_if
tff(fact_6282_the__inv__into__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,B),X5: B] : the_inv_into(A,B,A6,F3,X5) = the(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_wc(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A6),F3),X5)) ).

% the_inv_into_def
tff(fact_6283_map__add__left__comm,axiom,
    ! [B: $tType,A: $tType,A6: fun(A,option(B)),B5: fun(A,option(B)),C4: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,A6)),dom(A,B,B5)) = bot_bot(set(A)) )
     => ( map_add(A,B,A6,map_add(A,B,B5,C4)) = map_add(A,B,B5,map_add(A,B,A6,C4)) ) ) ).

% map_add_left_comm
tff(fact_6284_restrict__map__eq_I1_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A6: set(B),K2: B] :
      ( ( aa(B,option(A),restrict_map(B,A,M,A6),K2) = none(A) )
    <=> ~ pp(aa(set(B),bool,member(B,K2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),dom(B,A,M)),A6))) ) ).

% restrict_map_eq(1)
tff(fact_6285_map__card__eq__iff,axiom,
    ! [B: $tType,A: $tType,M6: fun(A,option(B)),X: A,Y: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M6)))
     => ( ( aa(set(A),nat,finite_card(A),dom(A,B,M6)) = aa(set(B),nat,finite_card(B),ran(A,B,M6)) )
       => ( pp(aa(set(A),bool,member(A,X),dom(A,B,M6)))
         => ( ( aa(A,option(B),M6,X) = aa(A,option(B),M6,Y) )
          <=> ( X = Y ) ) ) ) ) ).

% map_card_eq_iff
tff(fact_6286_finite__map__to__set,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),map_to_set(A,B,M)))
    <=> pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M))) ) ).

% finite_map_to_set
tff(fact_6287_map__to__set__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),map_to_set(A,B,M)) ).

% map_to_set_dom
tff(fact_6288_set__to__map__dom,axiom,
    ! [B: $tType,A: $tType,S: set(product_prod(A,B))] : dom(A,B,set_to_map(A,B,S)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S) ).

% set_to_map_dom
tff(fact_6289_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => pp(aa(set(fun(A,option(B))),bool,finite_finite2(fun(A,option(B))),aa(fun(fun(A,option(B)),bool),set(fun(A,option(B))),collect(fun(A,option(B))),aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_aie(set(A),fun(set(B),fun(fun(A,option(B)),bool)),A6),B5)))) ) ) ).

% finite_set_of_finite_maps
tff(fact_6290_card__map__to__set,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),map_to_set(A,B,M)) = aa(set(A),nat,finite_card(A),dom(A,B,M)) ).

% card_map_to_set
tff(fact_6291_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P2: fun(fun(A,option(B)),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M)))
     => ( pp(aa(fun(A,option(B)),bool,P2,aTP_Lamp_bp(A,option(B))))
       => ( ! [K: A,V3: B,M5: fun(A,option(B))] :
              ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M5)))
             => ( ~ pp(aa(set(A),bool,member(A,K),dom(A,B,M5)))
               => ( pp(aa(fun(A,option(B)),bool,P2,M5))
                 => pp(aa(fun(A,option(B)),bool,P2,fun_upd(A,option(B),M5,K,aa(B,option(B),some(B),V3)))) ) ) )
         => pp(aa(fun(A,option(B)),bool,P2,M)) ) ) ) ).

% finite_Map_induct
tff(fact_6292_ran__is__image,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,option(A))] : ran(B,A,M6) = aa(set(B),set(A),image2(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),M6)),dom(B,A,M6)) ).

% ran_is_image
tff(fact_6293_map__add__distinct__le,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [M: fun(A,option(B)),M8: fun(A,option(B)),N: fun(A,option(B)),N2: fun(A,option(B))] :
          ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M8))
         => ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),N),N2))
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M8)),dom(A,B,N2)) = bot_bot(set(A)) )
             => pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N)),map_add(A,B,M8,N2))) ) ) ) ) ).

% map_add_distinct_le
tff(fact_6294_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_aif(fun(A,option(B)),fun(A,product_prod(A,B)),M)),dom(A,B,M)) ).

% graph_eq_to_snd_dom
tff(fact_6295_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),X: B,B5: set(A)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(B),bool,member(B,X),aa(set(A),set(B),image2(A,B,F3),A6)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => pp(aa(set(A),bool,member(A,the_inv_into(A,B,A6,F3,X)),B5)) ) ) ) ).

% the_inv_into_into
tff(fact_6296_graph__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( graph(A,B,map_add(A,B,M1,M22)) = 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))),graph(A,B,M1)),graph(A,B,M22)) ) ) ).

% graph_map_add
tff(fact_6297_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
    <=> ? [V5: B] : F3 = fun_upd(A,option(B),aTP_Lamp_bp(A,option(B)),X,aa(B,option(B),some(B),V5)) ) ).

% dom_eq_singleton_conv
tff(fact_6298_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),M: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_aif(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).

% map_of_map_keys
tff(fact_6299_inj__on__map__the,axiom,
    ! [B: $tType,A: $tType,D5: set(A),M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D5),dom(A,B,M)))
     => ( inj_on(A,option(B),M,D5)
       => inj_on(A,B,aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),M),D5) ) ) ).

% inj_on_map_the
tff(fact_6300_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G3: fun(A,B),C4: set(A),B5: set(A),X: A] :
      ( inj_on(A,B,G3,C4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))
       => pp(aa(set(fun(B,A)),bool,member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_aig(fun(A,B),fun(set(A),fun(A,fun(B,A))),G3),C4),X)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6301_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),G3: fun(A,option(B)),A6: set(A)] : dom(A,B,override_on(A,option(B),F3,G3,A6)) = 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)),minus_minus(set(A)),dom(A,B,F3)),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aih(fun(A,option(B)),fun(set(A),fun(A,bool)),G3),A6)))),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aii(fun(A,option(B)),fun(set(A),fun(A,bool)),G3),A6))) ).

% dom_override_on
tff(fact_6302_Func__empty,axiom,
    ! [B: $tType,A: $tType,B5: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B5) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert(fun(A,B)),aTP_Lamp_aij(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_6303_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A15: set(B),B14: set(A),F22: fun(C,D),B23: set(C),A24: set(D)] :
      ( ( aa(set(B),set(A),image2(B,A,F1),A15) = B14 )
     => ( inj_on(C,D,F22,B23)
       => ( pp(aa(set(D),bool,aa(set(D),fun(set(D),bool),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F22),B23)),A24))
         => ( ( ( B23 = bot_bot(set(C)) )
             => ( A24 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B23,B14) = aa(set(fun(D,B)),set(fun(C,A)),image2(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B23,F1,F22)),bNF_Wellorder_Func(D,B,A24,A15)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_6304_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G3: fun(A,B),A24: set(A),A15: set(B),F1: fun(B,C),B14: set(C),F22: fun(D,A),B23: set(D)] :
      ( pp(aa(set(fun(A,B)),bool,member(fun(A,B),G3),bNF_Wellorder_Func(A,B,A24,A15)))
     => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F1),A15)),B14))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,F22),B23)),A24))
         => pp(aa(set(fun(D,C)),bool,member(fun(D,C),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B23,F1,F22),G3)),bNF_Wellorder_Func(D,C,B23,B14))) ) ) ) ).

% Func_map
tff(fact_6305_total__on__singleton,axiom,
    ! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),X),X)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_6306_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_aik(rat,fun(product_prod(int,int),bool),X)) ).

% quotient_of_def
tff(fact_6307_normalize__stable,axiom,
    ! [Q5: int,P3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q5))
     => ( algebr8660921524188924756oprime(int,P3,Q5)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q5)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q5) ) ) ) ).

% normalize_stable
tff(fact_6308_total__on__def,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( total_on(A,A6,R3)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A6))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),A6))
             => ( ( X4 != Xa3 )
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R3))
                  | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),R3)) ) ) ) ) ) ).

% total_on_def
tff(fact_6309_total__onI,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => ( pp(aa(set(A),bool,member(A,Y3),A6))
           => ( ( X3 != Y3 )
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
                | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),R3)) ) ) ) )
     => total_on(A,A6,R3) ) ).

% total_onI
tff(fact_6310_prod__coprime__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(B),A3: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => algebr8660921524188924756oprime(A,A3,aa(B,A,F3,I3)) )
         => algebr8660921524188924756oprime(A,A3,groups7121269368397514597t_prod(B,A,F3,A6)) ) ) ).

% prod_coprime_right
tff(fact_6311_prod__coprime__left,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(B),F3: fun(B,A),A3: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),A6))
             => algebr8660921524188924756oprime(A,aa(B,A,F3,I3),A3) )
         => algebr8660921524188924756oprime(A,groups7121269368397514597t_prod(B,A,F3,A6),A3) ) ) ).

% prod_coprime_left
tff(fact_6312_total__on__lex__prod,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(A,A)),B5: set(B),S2: set(product_prod(B,B))] :
      ( total_on(A,A6,R3)
     => ( total_on(B,B5,S2)
       => total_on(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),lex_prod(A,B,R3,S2)) ) ) ).

% total_on_lex_prod
tff(fact_6313_coprime__add__one__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),A3) ) ).

% coprime_add_one_left
tff(fact_6314_coprime__add__one__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% coprime_add_one_right
tff(fact_6315_Rat__cases,axiom,
    ! [Q5: rat] :
      ~ ! [A5: int,B4: int] :
          ( ( Q5 = aa(int,rat,aa(int,fun(int,rat),fract,A5),B4) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
           => ~ algebr8660921524188924756oprime(int,A5,B4) ) ) ).

% Rat_cases
tff(fact_6316_Rat__induct,axiom,
    ! [P2: fun(rat,bool),Q5: rat] :
      ( ! [A5: int,B4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
         => ( algebr8660921524188924756oprime(int,A5,B4)
           => pp(aa(rat,bool,P2,aa(int,rat,aa(int,fun(int,rat),fract,A5),B4))) ) )
     => pp(aa(rat,bool,P2,Q5)) ) ).

% Rat_induct
tff(fact_6317_Rat__cases__nonzero,axiom,
    ! [Q5: rat] :
      ( ! [A5: int,B4: int] :
          ( ( Q5 = aa(int,rat,aa(int,fun(int,rat),fract,A5),B4) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
           => ( ( A5 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A5,B4) ) ) )
     => ( Q5 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_6318_quotient__of__unique,axiom,
    ! [R3: rat] :
    ? [X3: product_prod(int,int)] :
      ( ( R3 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X3)),aa(product_prod(int,int),int,product_snd(int,int),X3)) )
      & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),X3)))
      & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X3),aa(product_prod(int,int),int,product_snd(int,int),X3))
      & ! [Y5: product_prod(int,int)] :
          ( ( ( R3 = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y5)),aa(product_prod(int,int),int,product_snd(int,int),Y5)) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y5)))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y5),aa(product_prod(int,int),int,product_snd(int,int),Y5)) )
         => ( Y5 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_6319_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),field_char_0_Rats(A)))
         => ~ ! [A5: int,B4: int] :
                ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
               => ( algebr8660921524188924756oprime(int,A5,B4)
                 => ( X != aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A5)),aa(int,A,ring_1_of_int(A),B4)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_6320_card__quotient__disjoint,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( inj_on(A,set(set(A)),aTP_Lamp_ail(set(product_prod(A,A)),fun(A,set(set(A))),R3),A6)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A6,R3)) = aa(set(A),nat,finite_card(A),A6) ) ) ) ).

% card_quotient_disjoint
tff(fact_6321_Rats__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),field_char_0_Rats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),field_char_0_Rats(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_add
tff(fact_6322_Ints__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ring_1_Ints(A)),field_char_0_Rats(A))) ) ).

% Ints_subset_Rats
tff(fact_6323_quotient__diff1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),A3: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_ail(set(product_prod(A,A)),fun(A,set(set(A))),R3),A6)
     => ( pp(aa(set(A),bool,member(A,A3),A6))
       => ( equiv_quotient(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))),R3) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),minus_minus(set(set(A))),equiv_quotient(A,A6,R3)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))),R3)) ) ) ) ).

% quotient_diff1
tff(fact_6324_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_6325_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),N) ) ).

% coprime_diff_one_left_nat
tff(fact_6326_finite__equiv__class,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
       => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
         => pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).

% finite_equiv_class
tff(fact_6327_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),A6: set(A),G3: fun(B,nat),B5: set(B)] :
      ( inj_on(A,nat,F3,A6)
     => ( inj_on(B,nat,G3,B5)
       => ( ! [A5: A,B4: B] :
              ( pp(aa(set(A),bool,member(A,A5),A6))
             => ( pp(aa(set(B),bool,member(B,B4),B5))
               => algebr8660921524188924756oprime(nat,aa(A,nat,F3,A5),aa(B,nat,G3,B4)) ) )
         => inj_on(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_aim(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F3),G3)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_6328_finite__quotient,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
       => pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A6,R3))) ) ) ).

% finite_quotient
tff(fact_6329_quotient__def,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] : equiv_quotient(A,A6,R3) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),image2(A,set(set(A)),aTP_Lamp_ain(set(product_prod(A,A)),fun(A,set(set(A))),R3)),A6)) ).

% quotient_def
tff(fact_6330_eq__f__restr__ss__eq,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(fun(A,option(B)),fun(A,option(B))),A6: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A6))))
     => ( ( A6 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A6),aa(set(A),set(A),uminus_uminus(set(A)),S2)) )
      <=> ( map_le(A,B,A6,aa(fun(A,option(B)),fun(A,option(B)),F3,A6))
          & ( S2 = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A6))),dom(A,B,A6)) ) ) ) ) ).

% eq_f_restr_ss_eq
tff(fact_6331_ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R3: set(product_prod(A,B)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
     => ( pp(aa(set(A),bool,member(A,A3),A6))
       => pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image(A,B,R3),A6))) ) ) ).

% ImageI
tff(fact_6332_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R3: set(product_prod(B,A)),A3: B] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B))))))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R3)) ) ).

% Image_singleton_iff
tff(fact_6333_pair__vimage__is__Image,axiom,
    ! [A: $tType,B: $tType,U: B,E6: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),aa(fun(A,product_prod(B,A)),fun(set(product_prod(B,A)),set(A)),vimage(A,product_prod(B,A)),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U)),E6) = aa(set(B),set(A),image(B,A,E6),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),U),bot_bot(set(B)))) ).

% pair_vimage_is_Image
tff(fact_6334_Image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(B,A)),B5: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),image(B,A,R3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_aio(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R3),B5)),A6)) ).

% Image_UN
tff(fact_6335_rtrancl__image__advance__rtrancl,axiom,
    ! [A: $tType,Q5: A,R2: set(product_prod(A,A)),Q0: set(A),X: A] :
      ( pp(aa(set(A),bool,member(A,Q5),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),X)),transitive_rtrancl(A,R2)))
       => pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0))) ) ) ).

% rtrancl_image_advance_rtrancl
tff(fact_6336_rtrancl__image__advance,axiom,
    ! [A: $tType,Q5: A,R2: set(product_prod(A,A)),Q0: set(A),X: A] :
      ( pp(aa(set(A),bool,member(A,Q5),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q5),X)),R2))
       => pp(aa(set(A),bool,member(A,X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),Q0))) ) ) ).

% rtrancl_image_advance
tff(fact_6337_ImageE,axiom,
    ! [A: $tType,B: $tType,B2: A,R3: set(product_prod(B,A)),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R3),A6)))
     => ~ ! [X3: B] :
            ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),B2)),R3))
           => ~ pp(aa(set(B),bool,member(B,X3),A6)) ) ) ).

% ImageE
tff(fact_6338_Image__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R3: set(product_prod(B,A)),A6: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R3),A6)))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A6))
          & pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),B2)),R3)) ) ) ).

% Image_iff
tff(fact_6339_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,R3: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,A3),A6))
     => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
       => pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image(A,B,R3),A6))) ) ) ).

% rev_ImageI
tff(fact_6340_Image__Un,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A6: set(B),B5: set(B)] : aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5)) = 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,R2),A6)),aa(set(B),set(A),image(B,A,R2),B5)) ).

% Image_Un
tff(fact_6341_Un__Image,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S: set(product_prod(B,A)),A6: set(B)] : aa(set(B),set(A),image(B,A,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))),R2),S)),A6) = 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,R2),A6)),aa(set(B),set(A),image(B,A,S),A6)) ).

% Un_Image
tff(fact_6342_Image__Int__subset,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A6: set(B),B5: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B5))),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,R2),A6)),aa(set(B),set(A),image(B,A,R2),B5)))) ).

% Image_Int_subset
tff(fact_6343_Image__closed__trancl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),X6)),X6))
     => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R3)),X6) = X6 ) ) ).

% Image_closed_trancl
tff(fact_6344_rtrancl__reachable__induct,axiom,
    ! [A: $tType,I5: set(A),INV: set(A),E6: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),I5),INV))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E6),INV)),INV))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),I5)),INV)) ) ) ).

% rtrancl_reachable_induct
tff(fact_6345_rtrancl__image__unfold__right,axiom,
    ! [A: $tType,E6: set(product_prod(A,A)),V: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E6),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),V))),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),V))) ).

% rtrancl_image_unfold_right
tff(fact_6346_Image__mono,axiom,
    ! [B: $tType,A: $tType,R6: set(product_prod(A,B)),R3: set(product_prod(A,B)),A13: set(A),A6: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R6),R3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A13),A6))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,R6),A13)),aa(set(A),set(B),image(A,B,R3),A6))) ) ) ).

% Image_mono
tff(fact_6347_map__leI,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ! [X3: A,V3: B] :
          ( ( aa(A,option(B),M1,X3) = aa(B,option(B),some(B),V3) )
         => ( aa(A,option(B),M22,X3) = aa(B,option(B),some(B),V3) ) )
     => map_le(A,B,M1,M22) ) ).

% map_leI
tff(fact_6348_map__leD,axiom,
    ! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),K2: A,V2: B] :
      ( map_le(A,B,M1,M22)
     => ( ( aa(A,option(B),M1,K2) = aa(B,option(B),some(B),V2) )
       => ( aa(A,option(B),M22,K2) = aa(B,option(B),some(B),V2) ) ) ) ).

% map_leD
tff(fact_6349_map__le__empty,axiom,
    ! [B: $tType,A: $tType,G3: fun(A,option(B))] : map_le(A,B,aTP_Lamp_bp(A,option(B)),G3) ).

% map_le_empty
tff(fact_6350_trancl__Image__unfold__left,axiom,
    ! [A: $tType,E6: set(product_prod(A,A)),S: set(A)] : aa(set(A),set(A),image(A,A,transitive_trancl(A,E6)),S) = aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),aa(set(A),set(A),image(A,A,E6),S)) ).

% trancl_Image_unfold_left
tff(fact_6351_trancl__Image__unfold__right,axiom,
    ! [A: $tType,E6: set(product_prod(A,A)),S: set(A)] : aa(set(A),set(A),image(A,A,transitive_trancl(A,E6)),S) = aa(set(A),set(A),image(A,A,E6),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),S)) ).

% trancl_Image_unfold_right
tff(fact_6352_finite__Image__subset,axiom,
    ! [A: $tType,B: $tType,A6: set(product_prod(B,A)),B5: set(B),C4: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image(B,A,A6),B5)))
     => ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),C4),A6))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image(B,A,C4),B5))) ) ) ).

% finite_Image_subset
tff(fact_6353_map__le__implies__dom__le,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( map_le(A,B,F3,G3)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),dom(A,B,F3)),dom(A,B,G3))) ) ).

% map_le_implies_dom_le
tff(fact_6354_Image__empty__rtrancl__Image__id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),V2: A] :
      ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A))) ) ) ).

% Image_empty_rtrancl_Image_id
tff(fact_6355_Image__empty__trancl__Image__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),V2: A] :
      ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),image(A,A,transitive_trancl(A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ).

% Image_empty_trancl_Image_empty
tff(fact_6356_reachable__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R8: set(product_prod(A,A)),X6: set(A),X11: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),R8))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),X11))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),X6)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R8)),X11))) ) ) ).

% reachable_mono
tff(fact_6357_Image__subset__snd__image,axiom,
    ! [A: $tType,B: $tType,A6: set(product_prod(B,A)),B5: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,A6),B5)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A6))) ).

% Image_subset_snd_image
tff(fact_6358_trancl__image__by__rtrancl,axiom,
    ! [A: $tType,E6: set(product_prod(A,A)),Vi: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),image(A,A,transitive_trancl(A,E6)),Vi)),Vi) = aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),Vi) ).

% trancl_image_by_rtrancl
tff(fact_6359_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),A3: B] : aa(set(B),set(A),image(B,A,R3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),bot_bot(set(B)))) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_at(set(product_prod(B,A)),fun(B,fun(A,bool)),R3),A3)) ).

% Image_singleton
tff(fact_6360_Image__subset,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B)),A6: set(A),B5: set(B),C4: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,R3),C4)),B5)) ) ).

% Image_subset
tff(fact_6361_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(B,A)),B5: fun(C,set(B)),A6: set(C)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A6)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_aio(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R3),B5)),A6)))) ).

% Image_INT_subset
tff(fact_6362_rtrancl__apply__insert,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,S: set(A)] : aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ).

% rtrancl_apply_insert
tff(fact_6363_map__le__imp__upd__le,axiom,
    ! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X: A,Y: B] :
      ( map_le(A,B,M1,M22)
     => map_le(A,B,fun_upd(A,option(B),M1,X,none(B)),fun_upd(A,option(B),M22,X,aa(B,option(B),some(B),Y))) ) ).

% map_le_imp_upd_le
tff(fact_6364_E__closed__restr__reach__cases,axiom,
    ! [A: $tType,U: A,V2: A,E6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),transitive_rtrancl(A,E6)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E6),R2)),R2))
       => ( ~ pp(aa(set(A),bool,member(A,V2),R2))
         => ~ ( ~ pp(aa(set(A),bool,member(A,U),R2))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),transitive_rtrancl(A,rel_restrict(A,E6,R2)))) ) ) ) ) ).

% E_closed_restr_reach_cases
tff(fact_6365_rel__restrict__tranclI,axiom,
    ! [A: $tType,X: A,Y: A,E6: set(product_prod(A,A)),R2: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,E6)))
     => ( ~ pp(aa(set(A),bool,member(A,X),R2))
       => ( ~ pp(aa(set(A),bool,member(A,Y),R2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E6),R2)),R2))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,rel_restrict(A,E6,R2)))) ) ) ) ) ).

% rel_restrict_tranclI
tff(fact_6366_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),B5: set(B)] : aa(set(B),set(A),image(B,A,R3),B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aip(set(product_prod(B,A)),fun(B,set(A)),R3)),B5)) ).

% Image_eq_UN
tff(fact_6367_Sigma__Image,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B5: fun(B,set(A)),X6: set(B)] : aa(set(B),set(A),image(B,A,product_Sigma(B,A,A6,B5)),X6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X6),A6))) ).

% Sigma_Image
tff(fact_6368_UN__Image,axiom,
    ! [B: $tType,A: $tType,C: $tType,X6: fun(C,set(product_prod(B,A))),I5: set(C),S: set(B)] : aa(set(B),set(A),image(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),X6),I5))),S) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_aiq(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),X6),S)),I5)) ).

% UN_Image
tff(fact_6369_finite__reachable__advance,axiom,
    ! [A: $tType,E6: set(product_prod(A,A)),V0: A,V2: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V0),bot_bot(set(A))))))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V0),V2)),transitive_rtrancl(A,E6)))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A)))))) ) ) ).

% finite_reachable_advance
tff(fact_6370_rtrancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V2: A,E6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),E6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,E6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))))) ) ).

% rtrancl_Image_advance_ss
tff(fact_6371_trancl__Image__advance__ss,axiom,
    ! [A: $tType,U: A,V2: A,E6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),E6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_trancl(A,E6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,transitive_trancl(A,E6)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))))) ) ).

% trancl_Image_advance_ss
tff(fact_6372_trancl__restrict__reachable,axiom,
    ! [A: $tType,U: A,V2: A,E6: set(product_prod(A,A)),S: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),transitive_trancl(A,E6)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,E6),S)),S))
       => ( pp(aa(set(A),bool,member(A,U),S))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V2)),transitive_trancl(A,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))),inf_inf(set(product_prod(A,A))),E6),product_Sigma(A,A,S,aTP_Lamp_abo(set(A),fun(A,set(A)),S)))))) ) ) ) ).

% trancl_restrict_reachable
tff(fact_6373_Image__fold,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R2))
     => ( aa(set(A),set(B),image(A,B,R2),S) = finite_fold(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_va(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R2) ) ) ).

% Image_fold
tff(fact_6374_eq__f__restr__conv,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(fun(A,option(B)),fun(A,option(B))),A6: fun(A,option(B))] :
      ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A6))))
        & ( A6 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A6),aa(set(A),set(A),uminus_uminus(set(A)),S2)) ) )
    <=> ( map_le(A,B,A6,aa(fun(A,option(B)),fun(A,option(B)),F3,A6))
        & ( S2 = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F3,A6))),dom(A,B,A6)) ) ) ) ).

% eq_f_restr_conv
tff(fact_6375_le__map__mmupd__not__dom,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K5: set(A),V2: B] : map_le(A,B,M,map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K5),dom(A,B,M)),V2)) ).

% le_map_mmupd_not_dom
tff(fact_6376_map__mmupd__update__less,axiom,
    ! [A: $tType,B: $tType,K5: set(A),K7: set(A),M: fun(A,option(B)),V2: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K5),K7))
     => map_le(A,B,map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K5),dom(A,B,M)),V2),map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K7),dom(A,B,M)),V2)) ) ).

% map_mmupd_update_less
tff(fact_6377_mmupd__notin__upd,axiom,
    ! [B: $tType,A: $tType,K2: A,K5: set(A),M: fun(A,option(B)),V2: B] :
      ( ~ pp(aa(set(A),bool,member(A,K2),K5))
     => ( aa(A,option(B),map_mmupd(A,B,M,K5,V2),K2) = aa(A,option(B),M,K2) ) ) ).

% mmupd_notin_upd
tff(fact_6378_map__mmupd__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),V2: B] : map_mmupd(A,B,M,bot_bot(set(A)),V2) = M ).

% map_mmupd_empty
tff(fact_6379_mmupd__in__upd,axiom,
    ! [A: $tType,B: $tType,K2: A,K5: set(A),M: fun(A,option(B)),V2: B] :
      ( pp(aa(set(A),bool,member(A,K2),K5))
     => ( aa(A,option(B),map_mmupd(A,B,M,K5,V2),K2) = aa(B,option(B),some(B),V2) ) ) ).

% mmupd_in_upd
tff(fact_6380_dom__mmupd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K5: set(A),V2: B] : dom(A,B,map_mmupd(A,B,M,K5,V2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,M)),K5) ).

% dom_mmupd
tff(fact_6381_map__mmupdE,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),K5: set(B),V2: A,K2: B,X: A] :
      ( ( aa(B,option(A),map_mmupd(B,A,M,K5,V2),K2) = aa(A,option(A),some(A),X) )
     => ( ( ~ pp(aa(set(B),bool,member(B,K2),K5))
         => ( aa(B,option(A),M,K2) != aa(A,option(A),some(A),X) ) )
       => ~ ( pp(aa(set(B),bool,member(B,K2),K5))
           => ( X != V2 ) ) ) ) ).

% map_mmupdE
tff(fact_6382_map__mmupd__def,axiom,
    ! [A: $tType,B: $tType,K2: B,K5: set(B),M: fun(B,option(A)),V2: A] :
      ( ( pp(aa(set(B),bool,member(B,K2),K5))
       => ( aa(B,option(A),map_mmupd(B,A,M,K5,V2),K2) = aa(A,option(A),some(A),V2) ) )
      & ( ~ pp(aa(set(B),bool,member(B,K2),K5))
       => ( aa(B,option(A),map_mmupd(B,A,M,K5,V2),K2) = aa(B,option(A),M,K2) ) ) ) ).

% map_mmupd_def
tff(fact_6383_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q2: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q2))
         => order_antimono(nat,A,aTP_Lamp_air(fun(A,A),fun(nat,A),Q2)) ) ) ).

% antimono_funpow
tff(fact_6384_sum__mset__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Y: B,A6: multiset(A)] : comm_m7189776963980413722m_mset(B,aa(multiset(A),multiset(B),image_mset(A,B,aTP_Lamp_ais(B,fun(A,B),Y)),A6)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(multiset(A),nat,size_size(multiset(A)),A6))),Y) ) ).

% sum_mset_constant
tff(fact_6385_Union__image__single__mset,axiom,
    ! [A: $tType,M: multiset(A)] : comm_m7189776963980413722m_mset(multiset(A),aa(multiset(A),multiset(multiset(A)),image_mset(A,multiset(A),aTP_Lamp_ait(A,multiset(A))),M)) = M ).

% Union_image_single_mset
tff(fact_6386_sum__mset_Oadd__mset,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [X: A,N7: multiset(A)] : comm_m7189776963980413722m_mset(A,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),N7)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),comm_m7189776963980413722m_mset(A,N7)) ) ).

% sum_mset.add_mset
tff(fact_6387_sum__mset_Ounion,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M6: multiset(A),N7: multiset(A)] : comm_m7189776963980413722m_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),comm_m7189776963980413722m_mset(A,M6)),comm_m7189776963980413722m_mset(A,N7)) ) ).

% sum_mset.union
tff(fact_6388_sum__mset_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_fu(B,A)),A6)) = zero_zero(A) ) ).

% sum_mset.neutral_const
tff(fact_6389_sum__mset_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),X: B,A6: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X),A6))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G3,X)),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),A6))) ) ).

% sum_mset.insert
tff(fact_6390_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_antimono(A,B,F3)
        <=> ! [X4: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y4)),aa(A,B,F3,X4))) ) ) ) ).

% antimono_def
tff(fact_6391_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3))) )
         => order_antimono(A,B,F3) ) ) ).

% antimonoI
tff(fact_6392_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoE
tff(fact_6393_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoD
tff(fact_6394_sum__mset_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,fun(C,A)),B5: multiset(C),A6: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_aiu(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),G3),B5)),A6)) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_aiv(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),G3),A6)),B5)) ) ).

% sum_mset.swap
tff(fact_6395_sum__mset_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),H: fun(B,A),A6: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fz(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A6)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),A6))),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,H),A6))) ) ).

% sum_mset.distrib
tff(fact_6396_sum__mset__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C3: A,F3: fun(B,A),M6: multiset(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F3),M6))) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gd(A,fun(fun(B,A),fun(B,A)),C3),F3)),M6)) ) ).

% sum_mset_distrib_left
tff(fact_6397_sum__mset__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),M6: multiset(B),C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F3),M6))),C3) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_gc(fun(B,A),fun(A,fun(B,A)),F3),C3)),M6)) ) ).

% sum_mset_distrib_right
tff(fact_6398_sum__mset__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & times(A)
        & semiring_0(B) )
     => ! [F3: fun(A,B),A6: multiset(A),G3: fun(C,B),B5: multiset(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),comm_m7189776963980413722m_mset(B,aa(multiset(A),multiset(B),image_mset(A,B,F3),A6))),comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,G3),B5))) = comm_m7189776963980413722m_mset(B,aa(multiset(A),multiset(B),image_mset(A,B,aa(multiset(C),fun(A,B),aa(fun(C,B),fun(multiset(C),fun(A,B)),aTP_Lamp_aix(fun(A,B),fun(fun(C,B),fun(multiset(C),fun(A,B))),F3),G3),B5)),A6)) ) ).

% sum_mset_product
tff(fact_6399_size__eq__sum__mset,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),M6) = comm_m7189776963980413722m_mset(nat,aa(multiset(A),multiset(nat),image_mset(A,nat,aTP_Lamp_mj(A,nat)),M6)) ).

% size_eq_sum_mset
tff(fact_6400_antimono__iff__le__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_antimono(nat,A,F3)
        <=> ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N3))),aa(nat,A,F3,N3))) ) ) ).

% antimono_iff_le_Suc
tff(fact_6401_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),A6: set(A),Z4: B,Y: B,A3: A] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,A6),Y))
         => ( pp(aa(set(A),bool,member(A,A3),A6))
           => ? [Y8: B] :
                ( ( Y = aa(B,B,aa(A,fun(B,B),F3,A3),Y8) )
                & pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),Y8)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_6402_Field__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),A3),B2)),R3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),field2(A,R3)) ).

% Field_insert
tff(fact_6403_Field__square,axiom,
    ! [A: $tType,X: set(A)] : field2(A,product_Sigma(A,A,X,aTP_Lamp_abo(set(A),fun(A,set(A)),X))) = X ).

% Field_square
tff(fact_6404_finite__Field__eq__finite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),field2(A,R2)))
    <=> pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2)) ) ).

% finite_Field_eq_finite
tff(fact_6405_Field__Un,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] : field2(A,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))),R3),S2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),field2(A,R3)),field2(A,S2)) ).

% Field_Un
tff(fact_6406_mono__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),field2(A,R3)),field2(A,S2))) ) ).

% mono_Field
tff(fact_6407_FieldI1,axiom,
    ! [A: $tType,I2: A,J2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2))
     => pp(aa(set(A),bool,member(A,I2),field2(A,R2))) ) ).

% FieldI1
tff(fact_6408_FieldI2,axiom,
    ! [A: $tType,I2: A,J2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2))
     => pp(aa(set(A),bool,member(A,J2),field2(A,R2))) ) ).

% FieldI2
tff(fact_6409_R__subset__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,field2(A,R2),aTP_Lamp_aiy(set(product_prod(A,A)),fun(A,set(A)),R2)))) ).

% R_subset_Field
tff(fact_6410_Restr__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : 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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,field2(A,R3),aTP_Lamp_aiy(set(product_prod(A,A)),fun(A,set(A)),R3))) = R3 ).

% Restr_Field
tff(fact_6411_comp__fun__commute__on_Ofold__graph__determ,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),A6: set(A),Z4: B,X: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,A6),X))
         => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,A6),Y))
           => ( Y = X ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_determ
tff(fact_6412_fst__in__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_fst(A,A)),R2)),field2(A,R2))) ).

% fst_in_Field
tff(fact_6413_snd__in__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_snd(A,A)),R2)),field2(A,R2))) ).

% snd_in_Field
tff(fact_6414_rel__restrict__Int__empty,axiom,
    ! [A: $tType,A6: set(A),R2: set(product_prod(A,A))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),field2(A,R2)) = bot_bot(set(A)) )
     => ( rel_restrict(A,R2,A6) = R2 ) ) ).

% rel_restrict_Int_empty
tff(fact_6415_Field__rel__restrict,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),field2(A,rel_restrict(A,R2,A6))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),field2(A,R2)),A6))) ).

% Field_rel_restrict
tff(fact_6416_Field__Restr__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))),A6)) ).

% Field_Restr_subset
tff(fact_6417_trancl__subset__Field2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),product_Sigma(A,A,field2(A,R3),aTP_Lamp_aiy(set(product_prod(A,A)),fun(A,set(A)),R3)))) ).

% trancl_subset_Field2
tff(fact_6418_Field__natLeq__on,axiom,
    ! [N: nat] : field2(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),N)))) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N)) ).

% Field_natLeq_on
tff(fact_6419_Total__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( total_on(A,field2(A,R3),R3)
     => total_on(A,field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% Total_Restr
tff(fact_6420_total__on__imp__Total__Restr,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( total_on(A,A6,R3)
     => total_on(A,field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% total_on_imp_Total_Restr
tff(fact_6421_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B,V2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),V2))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ~ ! [Y3: B] :
                  ( ( V2 = aa(B,B,aa(A,fun(B,B),F3,X),Y3) )
                 => ~ pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,A6),Y3)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_6422_comp__fun__commute__on_Ofold__equality,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),A6: set(A),Z4: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
       => ( pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,A6),Y))
         => ( finite_fold(A,B,F3,Z4,A6) = Y ) ) ) ) ).

% comp_fun_commute_on.fold_equality
tff(fact_6423_rtrancl__Image__in__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),V: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),V)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),field2(A,R2)),V))) ).

% rtrancl_Image_in_Field
tff(fact_6424_Finite__Set_Ofold__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,fun(B,B)),Z4: B] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),A6))
       => ( finite_fold(A,B,F3,Z4,A6) = the(B,finite_fold_graph(A,B,F3,Z4,A6)) ) )
      & ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
       => ( finite_fold(A,B,F3,Z4,A6) = Z4 ) ) ) ).

% Finite_Set.fold_def
tff(fact_6425_comp__fun__commute__on_Ofold__graph__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),A6: set(A),Z4: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => pp(aa(B,bool,finite_fold_graph(A,B,F3,Z4,A6),finite_fold(A,B,F3,Z4,A6))) ) ) ) ).

% comp_fun_commute_on.fold_graph_fold
tff(fact_6426_Total__subset__Id,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R3),R3)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),id2(A)))
       => ( ( R3 = bot_bot(set(product_prod(A,A))) )
          | ? [A5: A] : R3 = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),A5),A5)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_6427_Total__Id__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R3),R3)
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),id2(A)))
       => ( field2(A,R3) = field2(A,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))),minus_minus(set(product_prod(A,A))),R3),id2(A))) ) ) ) ).

% Total_Id_Field
tff(fact_6428_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_preorder_on(A,field2(A,R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R3)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3)) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_6429_bsqr__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R3) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_ajb(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R3)))) ).

% bsqr_def
tff(fact_6430_Field__bsqr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : field2(product_prod(A,A),bNF_Wellorder_bsqr(A,R3)) = product_Sigma(A,A,field2(A,R3),aTP_Lamp_aiy(set(product_prod(A,A)),fun(A,set(A)),R3)) ).

% Field_bsqr
tff(fact_6431_Preorder__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_preorder_on(A,field2(A,R3),R3)
     => order_preorder_on(A,field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% Preorder_Restr
tff(fact_6432_subset__Image__Image__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_preorder_on(A,field2(A,R3),R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),field2(A,R3)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),A6)),aa(set(A),set(A),image(A,A,R3),B5)))
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),A6))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),B5))
                    & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),R3)) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_6433_cofinal__def,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A6,R3)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),field2(A,R3)))
         => ? [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),A6))
              & ( X4 != Xa3 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R3)) ) ) ) ).

% cofinal_def
tff(fact_6434_linear__order__on__singleton,axiom,
    ! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),X),X)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_6435_Linear__order__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_679001287576687338der_on(A,field2(A,R3),R3)
     => order_679001287576687338der_on(A,field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% Linear_order_Restr
tff(fact_6436_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),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))),minus_minus(set(product_prod(A,A))),R3),id2(A)))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_6437_linear__order__on__Restr,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A] :
      ( order_679001287576687338der_on(A,A6,R3)
     => order_679001287576687338der_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),order_above(A,R3,X)),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_above(A,R3,X),aa(A,fun(A,set(A)),aTP_Lamp_ajc(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X)))) ) ).

% linear_order_on_Restr
tff(fact_6438_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R3),R3)
     => ( wf(A,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))),minus_minus(set(product_prod(A,A))),R3),id2(A)))
      <=> ! [A16: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A16),field2(A,R3)))
           => ( ( A16 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A16))
                  & ! [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),A16))
                     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R3)) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_6439_brk__rel__wf,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(product_prod(bool,A),brk_rel(A,A,R2)) ) ).

% brk_rel_wf
tff(fact_6440_wf__insert,axiom,
    ! [A: $tType,Y: A,X: A,R3: set(product_prod(A,A))] :
      ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),Y),X)),R3))
    <=> ( wf(A,R3)
        & ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3))) ) ) ).

% wf_insert
tff(fact_6441_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))) ) ).

% wf
tff(fact_6442_wf__if__measure,axiom,
    ! [A: $tType,P2: fun(A,bool),F3: fun(A,nat),G3: fun(A,A)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,aa(A,A,G3,X3))),aa(A,nat,F3,X3))) )
     => wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_ajd(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),P2),G3)))) ) ).

% wf_if_measure
tff(fact_6443_wf__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(A,A))] :
      ( wf(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),P3),R3))
       => wf(A,P3) ) ) ).

% wf_subset
tff(fact_6444_wf__relcomp__compatible,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),relcomp(A,A,A,S,R2)))
       => wf(A,relcomp(A,A,A,S,R2)) ) ) ).

% wf_relcomp_compatible
tff(fact_6445_wf__union__merge,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,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))),R2),S))
    <=> wf(A,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))),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))),relcomp(A,A,A,R2,R2)),relcomp(A,A,A,S,R2))),S)) ) ).

% wf_union_merge
tff(fact_6446_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ~ ? [F11: fun(nat,A)] :
          ! [I: nat] : pp(aa(set(product_prod(A,A)),bool,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,F11,aa(nat,nat,suc,I))),aa(nat,A,F11,I))),R3)) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_6447_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),F3: fun(nat,A)] :
      ( wf(A,R3)
     => ~ ! [K: nat] : pp(aa(set(product_prod(A,A)),bool,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,F3,aa(nat,nat,suc,K))),aa(nat,A,F3,K))),R3)) ) ).

% wf_no_infinite_down_chainE
tff(fact_6448_wf__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ! [P8: fun(A,bool)] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4)),R3))
                 => pp(aa(A,bool,P8,Y4)) )
             => pp(aa(A,bool,P8,X4)) )
         => ! [X_12: A] : pp(aa(A,bool,P8,X_12)) ) ) ).

% wf_def
tff(fact_6449_wfE__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Q2: set(A)] :
      ( wf(A,R2)
     => ( pp(aa(set(A),bool,member(A,X),Q2))
       => ~ ! [Z3: A] :
              ( pp(aa(set(A),bool,member(A,Z3),Q2))
             => ~ ! [Y5: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3)),R2))
                   => ~ pp(aa(set(A),bool,member(A,Y5),Q2)) ) ) ) ) ).

% wfE_min
tff(fact_6450_wfI__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X3: A,Q: set(A)] :
          ( pp(aa(set(A),bool,member(A,X3),Q))
         => ? [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),Q))
              & ! [Y3: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa)),R2))
                 => ~ pp(aa(set(A),bool,member(A,Y3),Q)) ) ) )
     => wf(A,R2) ) ).

% wfI_min
tff(fact_6451_wfUNIVI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ! [P: fun(A,bool),X3: A] :
          ( ! [Xa: A] :
              ( ! [Y3: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa)),R3))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,Xa)) )
         => pp(aa(A,bool,P,X3)) )
     => wf(A,R3) ) ).

% wfUNIVI
tff(fact_6452_wf__asym,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R3))
       => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R3)) ) ) ).

% wf_asym
tff(fact_6453_wf__induct,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( wf(A,R3)
     => ( ! [X3: A] :
            ( ! [Y5: A] :
                ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3)),R3))
               => pp(aa(A,bool,P2,Y5)) )
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,A3)) ) ) ).

% wf_induct
tff(fact_6454_wf__irrefl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( wf(A,R3)
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R3)) ) ).

% wf_irrefl
tff(fact_6455_wf__not__sym,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R3))
       => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R3)) ) ) ).

% wf_not_sym
tff(fact_6456_wf__not__refl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( wf(A,R3)
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R3)) ) ).

% wf_not_refl
tff(fact_6457_wf__eq__minimal,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ! [Q9: set(A)] :
          ( ? [X4: A] : pp(aa(set(A),bool,member(A,X4),Q9))
         => ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Q9))
              & ! [Y4: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4)),R3))
                 => ~ pp(aa(set(A),bool,member(A,Y4),Q9)) ) ) ) ) ).

% wf_eq_minimal
tff(fact_6458_wf__induct__rule,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( wf(A,R3)
     => ( ! [X3: A] :
            ( ! [Y5: A] :
                ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3)),R3))
               => pp(aa(A,bool,P2,Y5)) )
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,A3)) ) ) ).

% wf_induct_rule
tff(fact_6459_wfE__min_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Q2: set(A)] :
      ( wf(A,R2)
     => ( ( Q2 != bot_bot(set(A)) )
       => ~ ! [Z3: A] :
              ( pp(aa(set(A),bool,member(A,Z3),Q2))
             => ~ ! [Y5: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3)),R2))
                   => ~ pp(aa(set(A),bool,member(A,Y5),Q2)) ) ) ) ) ).

% wfE_min'
tff(fact_6460_wf__no__loop,axiom,
    ! [B: $tType,R2: set(product_prod(B,B))] :
      ( ( relcomp(B,B,B,R2,R2) = bot_bot(set(product_prod(B,B))) )
     => wf(B,R2) ) ).

% wf_no_loop
tff(fact_6461_wf__less,axiom,
    wf(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat)))) ).

% wf_less
tff(fact_6462_wf__bounded__measure,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Ub: fun(A,nat),F3: fun(A,nat)] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A5)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,B4)),aa(A,nat,Ub,A5)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,A5)),aa(A,nat,F3,B4))) ) )
     => wf(A,R3) ) ).

% wf_bounded_measure
tff(fact_6463_wfI__pf,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [A11: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),aa(set(A),set(A),image(A,A,R2),A11)))
         => ( A11 = bot_bot(set(A)) ) )
     => wf(A,R2) ) ).

% wfI_pf
tff(fact_6464_wfE__pf,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A6: set(A)] :
      ( wf(A,R2)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),image(A,A,R2),A6)))
       => ( A6 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_6465_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),P2: fun(B,bool),K2: B,M: fun(B,A)] :
      ( wf(A,R3)
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),transitive_trancl(A,R3)))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),transitive_rtrancl(A,R3))) )
       => ( pp(aa(B,bool,P2,K2))
         => ? [X3: B] :
              ( pp(aa(B,bool,P2,X3))
              & ! [Y5: B] :
                  ( pp(aa(B,bool,P2,Y5))
                 => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M,X3)),aa(B,A,M,Y5))),transitive_rtrancl(A,R3))) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_6466_wf__union__compatible,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( wf(A,S)
       => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),R2))
         => wf(A,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))),R2),S)) ) ) ) ).

% wf_union_compatible
tff(fact_6467_wfI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))))
     => ( ! [X3: A,P: fun(A,bool)] :
            ( ! [Xa: A] :
                ( ! [Y3: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa)),R3))
                   => pp(aa(A,bool,P,Y3)) )
               => pp(aa(A,bool,P,Xa)) )
           => ( pp(aa(set(A),bool,member(A,X3),A6))
             => ( pp(aa(set(A),bool,member(A,X3),B5))
               => pp(aa(A,bool,P,X3)) ) ) )
       => wf(A,R3) ) ) ).

% wfI
tff(fact_6468_wf__eq__minimal2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( wf(A,R3)
    <=> ! [A16: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A16),field2(A,R3)))
            & ( A16 != bot_bot(set(A)) ) )
         => ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A16))
              & ! [Xa3: A] :
                  ( pp(aa(set(A),bool,member(A,Xa3),A16))
                 => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),R3)) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_6469_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),Ub: fun(A,set(B)),F3: fun(A,set(B))] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R3))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Ub,A5)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B4)),aa(A,set(B),Ub,A5)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,B4)),aa(A,set(B),Ub,A5)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F3,A5)),aa(A,set(B),F3,B4))) ) )
     => wf(A,R3) ) ).

% wf_bounded_set
tff(fact_6470_qc__wf__relto__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),relcomp(A,A,A,transitive_rtrancl(A,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))),R2),S)),R2)))
     => ( wf(A,relcomp(A,A,A,transitive_rtrancl(A,S),relcomp(A,A,A,R2,transitive_rtrancl(A,S))))
      <=> wf(A,R2) ) ) ).

% qc_wf_relto_iff
tff(fact_6471_above__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : order_above(A,R3,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),A3)) ).

% above_def
tff(fact_6472_wf__bounded__supset,axiom,
    ! [A: $tType,S: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_aje(set(A),fun(set(A),fun(set(A),bool)),S)))) ) ).

% wf_bounded_supset
tff(fact_6473_finite__subset__wf,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_ajf(set(A),fun(set(A),fun(set(A),bool)),A6)))) ) ).

% finite_subset_wf
tff(fact_6474_reduction__pairI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,S)),R2))
       => fun_reduction_pair(A,aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R2),S)) ) ) ).

% reduction_pairI
tff(fact_6475_reduction__pair__lemma,axiom,
    ! [A: $tType,P2: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( fun_reduction_pair(A,P2)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2)))
       => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P2)))
         => ( wf(A,S)
           => wf(A,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))),R2),S)) ) ) ) ) ).

% reduction_pair_lemma
tff(fact_6476_reduction__pair__def,axiom,
    ! [A: $tType,P2: product_prod(set(product_prod(A,A)),set(product_prod(A,A)))] :
      ( fun_reduction_pair(A,P2)
    <=> ( wf(A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2))
        & pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P2))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2))) ) ) ).

% reduction_pair_def
tff(fact_6477_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P2: fun(fun(A,B),fun(A,fun(B,bool)))] :
      ( wf(A,R2)
     => ( ! [F: fun(A,B),G2: fun(A,B),X3: A,R: B] :
            ( ! [Z6: A] :
                ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),X3)),R2))
               => ( aa(A,B,F,Z6) = aa(A,B,G2,Z6) ) )
           => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),X3),R))
            <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,G2),X3),R)) ) )
       => ( ! [X3: A,F: fun(A,B)] :
              ( ! [Y5: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3)),R2))
                 => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),Y5),aa(A,B,F,Y5))) )
             => ? [X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),X3),X_13)) )
         => ? [F: fun(A,B)] :
            ! [X5: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),X5),aa(A,B,F,X5))) ) ) ) ).

% dependent_wf_choice
tff(fact_6478_rp__inv__image__rp,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),F3: fun(B,A)] :
      ( fun_reduction_pair(A,P2)
     => fun_reduction_pair(B,aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),fun_rp_inv_image(A,B),P2),F3)) ) ).

% rp_inv_image_rp
tff(fact_6479_dependent__wellorder__choice,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(fun(A,B),fun(A,fun(B,bool)))] :
          ( ! [R: B,F: fun(A,B),G2: fun(A,B),X3: A] :
              ( ! [Y5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X3))
                 => ( aa(A,B,F,Y5) = aa(A,B,G2,Y5) ) )
             => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),X3),R))
              <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,G2),X3),R)) ) )
         => ( ! [X3: A,F: fun(A,B)] :
                ( ! [Y5: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X3))
                   => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),Y5),aa(A,B,F,Y5))) )
               => ? [X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),X3),X_13)) )
           => ? [F: fun(A,B)] :
              ! [X5: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F),X5),aa(A,B,F,X5))) ) ) ) ).

% dependent_wellorder_choice
tff(fact_6480_chains__extend,axiom,
    ! [A: $tType,C3: set(set(A)),S: set(set(A)),Z4: set(A)] :
      ( pp(aa(set(set(set(A))),bool,member(set(set(A)),C3),chains2(A,S)))
     => ( pp(aa(set(set(A)),bool,member(set(A),Z4),S))
       => ( ! [X3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X3),C3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Z4)) )
         => pp(aa(set(set(set(A))),bool,member(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))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Z4),bot_bot(set(set(A))))),C3)),chains2(A,S))) ) ) ) ).

% chains_extend
tff(fact_6481_rp__inv__image__def,axiom,
    ! [B: $tType,A: $tType] : fun_rp_inv_image(A,B) = aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_ajg(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))))) ).

% rp_inv_image_def
tff(fact_6482_in__inv__image,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R3: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),inv_image(B,A,R3,F3)))
    <=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,X)),aa(A,B,F3,Y))),R3)) ) ).

% in_inv_image
tff(fact_6483_chainsD,axiom,
    ! [A: $tType,C3: set(set(A)),S: set(set(A)),X: set(A),Y: set(A)] :
      ( pp(aa(set(set(set(A))),bool,member(set(set(A)),C3),chains2(A,S)))
     => ( pp(aa(set(set(A)),bool,member(set(A),X),C3))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y),C3))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X),Y))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y),X)) ) ) ) ) ).

% chainsD
tff(fact_6484_Zorn__Lemma2,axiom,
    ! [A: $tType,A6: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,member(set(set(A)),X3),chains2(A,A6)))
         => ? [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa),A6))
              & ! [Xb3: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),Xb3),X3))
                 => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xb3),Xa)) ) ) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A6))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa),A6))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa))
               => ( Xa = X3 ) ) ) ) ) ).

% Zorn_Lemma2
tff(fact_6485_chainsD2,axiom,
    ! [A: $tType,C3: set(set(A)),S: set(set(A))] :
      ( pp(aa(set(set(set(A))),bool,member(set(set(A)),C3),chains2(A,S)))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C3),S)) ) ).

% chainsD2
tff(fact_6486_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(B,B)),F3: fun(A,B)] : inv_image(B,A,R3,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ajh(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),R3),F3))) ).

% inv_image_def
tff(fact_6487_Zorn__Lemma,axiom,
    ! [A: $tType,A6: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( pp(aa(set(set(set(A))),bool,member(set(set(A)),X3),chains2(A,A6)))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X3)),A6)) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A6))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa),A6))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa))
               => ( Xa = X3 ) ) ) ) ) ).

% Zorn_Lemma
tff(fact_6488_chains__def,axiom,
    ! [A: $tType,A6: set(set(A))] : chains2(A,A6) = aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_aji(set(set(A)),fun(set(set(A)),bool),A6)) ).

% chains_def
tff(fact_6489_lenlex__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R3)),aTP_Lamp_ajj(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_6490_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),less_than))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ).

% less_than_iff
tff(fact_6491_chain__subset__def,axiom,
    ! [A: $tType,C4: set(set(A))] :
      ( chain_subset(A,C4)
    <=> ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),C4))
         => ! [Xa3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa3),C4))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa3))
                | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa3),X4)) ) ) ) ) ).

% chain_subset_def
tff(fact_6492_mlex__prod__def,axiom,
    ! [A: $tType,F3: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F3,R2) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R2),aTP_Lamp_ajk(fun(A,nat),fun(A,product_prod(nat,A)),F3)) ).

% mlex_prod_def
tff(fact_6493_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat,S2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(A,set(A),set_ord_lessThan(A),S2)))))
           => ( infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(A,set(A),set_ord_lessThan(A),S2)),N) = infini527867602293511546merate(A,S,N) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_6494_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R2),A3),B2))
    <=> ? [Z2: list(product_prod(A,B))] :
          ( pp(aa(set(list(product_prod(A,B))),bool,member(list(product_prod(A,B)),Z2),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ajl(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R2))))
          & ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z2) = A3 )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z2) = B2 ) ) ) ).

% list.in_rel
tff(fact_6495_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M: nat,N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% enumerate_mono_iff
tff(fact_6496_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M: nat,N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(set(A),nat,finite_card(A),S)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N)))
              <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_6497_list_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,list_all2(A,B,R2),fequal(bool)),aTP_Lamp_rf(list(A),bool)),aTP_Lamp_ra(list(B),bool))) ).

% list.disc_transfer(2)
tff(fact_6498_list_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(list(B),bool),bool,aa(fun(list(A),bool),fun(fun(list(B),bool),bool),bNF_rel_fun(list(A),list(B),bool,bool,list_all2(A,B,R2),fequal(bool)),aTP_Lamp_ajm(list(A),bool)),aTP_Lamp_ajn(list(B),bool))) ).

% list.disc_transfer(1)
tff(fact_6499_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra2))
     => pp(aa(fun(list(A),fun(list(B),bool)),bool,aa(fun(list(A),fun(list(B),bool)),fun(fun(list(A),fun(list(B),bool)),bool),ord_less_eq(fun(list(A),fun(list(B),bool))),list_all2(A,B,R2)),list_all2(A,B,Ra2))) ) ).

% list.rel_mono
tff(fact_6500_Misc_Olist__all2__induct,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),L: list(A),L4: list(B),Q2: fun(list(A),fun(list(B),bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),L),L4))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q2,nil(A)),nil(B)))
       => ( ! [X3: A,X7: B,Ls: list(A),Ls3: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X3),X7))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Ls),Ls3))
               => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q2,Ls),Ls3))
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ls)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X7),Ls3))) ) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q2,L),L4)) ) ) ) ).

% Misc.list_all2_induct
tff(fact_6501_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: fun(A,fun(B,bool)),As: list(A),F3: fun(C,B),Bs: list(C)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),As),aa(list(C),list(B),map(C,B,F3),Bs)))
    <=> pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),list_all2(A,C,aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_ajo(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P2),F3)),As),Bs)) ) ).

% list_all2_map2
tff(fact_6502_list__all2__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P2: fun(A,fun(B,bool)),F3: fun(C,A),As: list(C),Bs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),aa(list(C),list(A),map(C,A,F3),As)),Bs))
    <=> pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),list_all2(C,B,aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_ajp(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P2),F3)),As),Bs)) ) ).

% list_all2_map1
tff(fact_6503_list_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: list(A),Y: list(B)] :
      ( pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),list_all2(C,B,Sb),aa(list(A),list(C),map(A,C,I2),X)),Y))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_abg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% list.rel_map(1)
tff(fact_6504_list_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: list(A),G3: fun(B,C),Y: list(B)] :
      ( pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),list_all2(A,C,Sa),X),aa(list(B),list(C),map(B,C,G3),Y)))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_abf(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G3)),X),Y)) ) ).

% list.rel_map(2)
tff(fact_6505_le__enumerate,axiom,
    ! [S: set(nat),N: nat] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),infini527867602293511546merate(nat,S,N))) ) ).

% le_enumerate
tff(fact_6506_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys2),I))) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_6507_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P2: fun(A,fun(B,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),A3)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,A3),N4)),aa(nat,B,nth(B,B2),N4))) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),A3),B2)) ) ) ).

% list_all2_all_nthI
tff(fact_6508_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P3),aa(list(B),nat,size_size(list(B)),Ys2)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3))) ) ) ).

% list_all2_nthD2
tff(fact_6509_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P3),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3))) ) ) ).

% list_all2_nthD
tff(fact_6510_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,N)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N)))) ) ) ).

% enumerate_step
tff(fact_6511_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,N: nat,S: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N))) ) ) ) ).

% enumerate_mono
tff(fact_6512_product__lists__set,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_ajr(list(list(A)),fun(list(A),bool),Xss)) ).

% product_lists_set
tff(fact_6513_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X6)))
             => ( infini527867602293511546merate(A,X6,I3) = infini527867602293511546merate(A,Y6,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite2(A),X6))
           => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
             => ( ( aa(set(A),nat,finite_card(A),X6) = aa(set(A),nat,finite_card(A),Y6) )
               => ( X6 = Y6 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_6514_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),S2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(set(A),bool,member(A,S2),S))
           => ? [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(set(A),nat,finite_card(A),S)))
                & ( infini527867602293511546merate(A,S,N4) = S2 ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_6515_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S)))
           => pp(aa(set(A),bool,member(A,infini527867602293511546merate(A,S,N)),S)) ) ) ) ).

% finite_enumerate_in_set
tff(fact_6516_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A6: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A6,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A6,bNF_rel_fun(A,B,A,B,A6,A6)),plus_plus(A)),plus_plus(B)))
           => pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A6),A6),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B))) ) ) ) ).

% sum_list_transfer
tff(fact_6517_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,N: nat,S: set(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( pp(aa(set(A),bool,finite_finite2(A),S))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(A),nat,finite_card(A),S)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,M)),infini527867602293511546merate(A,S,N))) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_6518_finite__le__enumerate,axiom,
    ! [S: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(nat),nat,finite_card(nat),S)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),infini527867602293511546merate(nat,S,N))) ) ) ).

% finite_le_enumerate
tff(fact_6519_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A6: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A6,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A6,bNF_rel_fun(A,B,A,B,A6,A6)),plus_plus(A)),plus_plus(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A6,bNF_rel_fun(A,B,A,B,A6,A6)),times_times(A)),times_times(B)))
             => pp(aa(fun(fun(D,B),fun(B,fun(list(D),B))),bool,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B5,A6),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A6,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B5),A6))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).

% horner_sum_transfer
tff(fact_6520_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S,N)),infini527867602293511546merate(A,S,aa(nat,nat,suc,N)))) ) ) ) ).

% finite_enumerate_step
tff(fact_6521_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X6)))
             => ( infini527867602293511546merate(A,X6,I3) = infini527867602293511546merate(A,Y6,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite2(A),X6))
           => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X6)),aa(set(A),nat,finite_card(A),Y6)))
               => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),Y6)) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_6522_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S)))
           => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_ajs(set(A),fun(nat,fun(A,bool)),S),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_6523_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] : infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),ord_Least(A,aTP_Lamp_ajt(set(A),fun(A,bool),S))),bot_bot(set(A)))),N) ) ).

% enumerate_Suc
tff(fact_6524_Least__eq__0,axiom,
    ! [P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ord_Least(nat,P2) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_6525_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K2: A,P2: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),ord_Least(A,P2)))
         => ~ pp(aa(A,bool,P2,K2)) ) ) ).

% not_less_Least
tff(fact_6526_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P2,X3))
               => pp(aa(A,bool,Q2,X3)) )
           => pp(aa(A,bool,Q2,ord_Least(A,P2))) ) ) ) ).

% LeastI2_ex
tff(fact_6527_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
         => pp(aa(A,bool,P2,ord_Least(A,P2))) ) ) ).

% LeastI_ex
tff(fact_6528_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),A3: A,Q2: fun(A,bool)] :
          ( pp(aa(A,bool,P2,A3))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P2,X3))
               => pp(aa(A,bool,Q2,X3)) )
           => pp(aa(A,bool,Q2,ord_Least(A,P2))) ) ) ) ).

% LeastI2
tff(fact_6529_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),K2: A] :
          ( pp(aa(A,bool,P2,K2))
         => pp(aa(A,bool,P2,ord_Least(A,P2))) ) ) ).

% LeastI
tff(fact_6530_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),K2: A] :
          ( pp(aa(A,bool,P2,K2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P2)),K2)) ) ) ).

% Least_le
tff(fact_6531_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),Q2: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
         => ( ! [A5: A] :
                ( pp(aa(A,bool,P2,A5))
               => ( ! [B10: A] :
                      ( pp(aa(A,bool,P2,B10))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B10)) )
                 => pp(aa(A,bool,Q2,A5)) ) )
           => pp(aa(A,bool,Q2,ord_Least(A,P2))) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_6532_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),A3: A,Q2: fun(A,bool)] :
          ( pp(aa(A,bool,P2,A3))
         => ( ! [A5: A] :
                ( pp(aa(A,bool,P2,A5))
               => ( ! [B10: A] :
                      ( pp(aa(A,bool,P2,B10))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B10)) )
                 => pp(aa(A,bool,Q2,A5)) ) )
           => pp(aa(A,bool,Q2,ord_Least(A,P2))) ) ) ) ).

% LeastI2_wellorder
tff(fact_6533_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( ord_Least(A,P2) = X ) ) ) ) ).

% Least_equality
tff(fact_6534_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A,Q2: fun(A,bool)] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P2,X3))
                 => ( ! [Y5: A] :
                        ( pp(aa(A,bool,P2,Y5))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y5)) )
                   => pp(aa(A,bool,Q2,X3)) ) )
             => pp(aa(A,bool,Q2,ord_Least(A,P2))) ) ) ) ) ).

% LeastI2_order
tff(fact_6535_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),Z4: A] :
          ( ? [X5: A] :
              ( pp(aa(A,bool,P2,X5))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P2,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P2,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Ya2)) ) )
                 => ( Y3 = X5 ) ) )
         => ( pp(aa(A,bool,P2,Z4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P2)),Z4)) ) ) ) ).

% Least1_le
tff(fact_6536_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool)] :
          ( ? [X5: A] :
              ( pp(aa(A,bool,P2,X5))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P2,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P2,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Ya2)) ) )
                 => ( Y3 = X5 ) ) )
         => pp(aa(A,bool,P2,ord_Least(A,P2))) ) ) ).

% Least1I
tff(fact_6537_Inf__nat__def,axiom,
    ! [X6: set(nat)] : aa(set(nat),nat,complete_Inf_Inf(nat),X6) = ord_Least(nat,aTP_Lamp_aju(set(nat),fun(nat,bool),X6)) ).

% Inf_nat_def
tff(fact_6538_Least__Suc2,axiom,
    ! [P2: fun(nat,bool),N: nat,Q2: fun(nat,bool),M: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( pp(aa(nat,bool,Q2,M))
       => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
         => ( ! [K: nat] :
                ( pp(aa(nat,bool,P2,aa(nat,nat,suc,K)))
              <=> pp(aa(nat,bool,Q2,K)) )
           => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,Q2)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_6539_Least__Suc,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
       => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ajv(fun(nat,bool),fun(nat,bool),P2))) ) ) ) ).

% Least_Suc
tff(fact_6540_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool)] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P2)))
         => ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
           => ( ord_Least(A,P2) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),P2)) ) ) ) ) ).

% Least_Min
tff(fact_6541_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),P2: fun(A,bool)] : abort_Bleast(A,S,P2) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajw(set(A),fun(fun(A,bool),fun(A,bool)),S),P2)) ) ).

% abort_Bleast_def
tff(fact_6542_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S: set(A),P2: fun(A,bool)] : bleast(A,S,P2) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajw(set(A),fun(fun(A,bool),fun(A,bool)),S),P2)) ) ).

% Bleast_def
tff(fact_6543_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] : infini527867602293511546merate(A,S,zero_zero(nat)) = ord_Least(A,aTP_Lamp_ajt(set(A),fun(A,bool),S)) ) ).

% enumerate_0
tff(fact_6544_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),S: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( ? [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),S))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,member(A,Xa4),S))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa4)) ) )
           => ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_ajx(fun(A,B),fun(set(A),fun(B,bool)),F3),S)) = aa(A,B,F3,ord_Least(A,aTP_Lamp_ajy(set(A),fun(A,bool),S))) ) ) ) ) ).

% Least_mono
tff(fact_6545_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S))
         => ( infini527867602293511546merate(A,S,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_ajs(set(A),fun(nat,fun(A,bool)),S),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_6546_finite__refines__card__le,axiom,
    ! [A: $tType,A6: set(A),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A6,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
       => ( equiv_equiv(A,A6,R2)
         => ( equiv_equiv(A,A6,S)
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A6,S))),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A6,R2)))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_6547_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F3),Xs),bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_6548_in__quotient__imp__subset,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A6)) ) ) ).

% in_quotient_imp_subset
tff(fact_6549_quotient__eqI,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A),Y6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y6),equiv_quotient(A,A6,R3)))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( pp(aa(set(A),bool,member(A,Y),Y6))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_6550_quotient__eq__iff,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A),Y6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y6),equiv_quotient(A,A6,R3)))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( pp(aa(set(A),bool,member(A,Y),Y6))
             => ( ( X6 = Y6 )
              <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_6551_in__quotient__imp__closed,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => ( pp(aa(set(A),bool,member(A,X),X6))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
           => pp(aa(set(A),bool,member(A,Y),X6)) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_6552_foldl__conv__fold,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),S2: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),S2),Xs) = aa(A,A,fold(B,A,aTP_Lamp_sm(fun(A,fun(B,A)),fun(B,fun(A,A)),F3),Xs),S2) ).

% foldl_conv_fold
tff(fact_6553_fold__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P2: fun(B,bool),Xs: list(B)] : fold(B,A,F3,filter2(B,P2,Xs)) = fold(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_afr(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F3),P2),Xs) ).

% fold_filter
tff(fact_6554_equiv__type,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( equiv_equiv(A,A6,R3)
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% equiv_type
tff(fact_6555_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A6: set(A)] : 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)),A6) = aa(set(A),set(A),fold(A,set(A),insert(A),Xs),A6) ).

% union_set_fold
tff(fact_6556_fold__plus__sum__list__rev,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : fold(A,A,plus_plus(A),Xs) = aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),rev(A,Xs))) ) ).

% fold_plus_sum_list_rev
tff(fact_6557_finite__refines__finite,axiom,
    ! [A: $tType,A6: set(A),R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A6,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
       => ( equiv_equiv(A,A6,R2)
         => ( equiv_equiv(A,A6,S)
           => pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A6,S))) ) ) ) ) ).

% finite_refines_finite
tff(fact_6558_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_pk(A,A)),Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_6559_Sup__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,sup_sup(A),Xs),bot_bot(A)) ) ).

% Sup_set_fold
tff(fact_6560_equiv__class__eq__iff,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
      <=> ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) )
          & pp(aa(set(A),bool,member(A,X),A6))
          & pp(aa(set(A),bool,member(A,Y),A6)) ) ) ) ).

% equiv_class_eq_iff
tff(fact_6561_eq__equiv__class__iff,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(A),bool,member(A,X),A6))
       => ( pp(aa(set(A),bool,member(A,Y),A6))
         => ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_6562_equiv__class__eq,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_6563_eq__equiv__class,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,A6: set(A)] :
      ( ( aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A6,R3)
       => ( pp(aa(set(A),bool,member(A,B2),A6))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ).

% eq_equiv_class
tff(fact_6564_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,sup_sup(A),Xs),X) ) ).

% Sup_fin.set_eq_fold
tff(fact_6565_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(A),bool,member(A,X),A6))
       => ( pp(aa(set(A),bool,member(A,Y),A6))
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),R3) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))),R3) )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_6566_refines__equiv__class__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A)),A6: set(A),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( equiv_equiv(A,A6,R2)
       => ( equiv_equiv(A,A6,S)
         => ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_6567_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A)),A6: set(A),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( equiv_equiv(A,A6,R2)
       => ( equiv_equiv(A,A6,S)
         => ( aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_6568_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite673082921795544331dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S))
       => ( finite_fold(A,B,F3,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,Xs),Y) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_6569_refines__equiv__image__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S))
     => ( equiv_equiv(A,A6,R2)
       => ( equiv_equiv(A,A6,S)
         => ( aa(set(set(A)),set(set(A)),image2(set(A),set(A),image(A,A,S)),equiv_quotient(A,A6,R2)) = equiv_quotient(A,A6,S) ) ) ) ) ).

% refines_equiv_image_eq
tff(fact_6570_equiv__class__subset,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))) ) ) ).

% equiv_class_subset
tff(fact_6571_subset__equiv__class,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),B2: A,A3: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))))
       => ( pp(aa(set(A),bool,member(A,B2),A6))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ).

% subset_equiv_class
tff(fact_6572_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A,A3: A,B2: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(A),bool,member(A,X),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,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ).

% equiv_class_nondisjoint
tff(fact_6573_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),X6))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_6574_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S))
       => ( finite_fold(A,B,F3,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,aa(list(A),list(A),remdups(A),Xs)),Y) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_6575_UN__equiv__class2,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),A1: A,A22: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F3)
         => ( pp(aa(set(A),bool,member(A,A1),A15))
           => ( pp(aa(set(B),bool,member(B,A22),A24))
             => ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_ajz(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F3),A22)),aa(set(A),set(A),image(A,A,R12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F3,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_6576_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(A,A)),F3: fun(A,set(B)),A3: A] :
      ( equiv_equiv(A,A6,R3)
     => ( equiv_congruent(A,set(B),R3,F3)
       => ( pp(aa(set(A),bool,member(A,A3),A6))
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))))) = aa(A,set(B),F3,A3) ) ) ) ) ).

% UN_equiv_class
tff(fact_6577_congruentD,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,B),Y: A,Z4: A] :
      ( equiv_congruent(A,B,R3,F3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),R3))
       => ( aa(A,B,F3,Y) = aa(A,B,F3,Z4) ) ) ) ).

% congruentD
tff(fact_6578_congruentI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,B)] :
      ( ! [Y3: A,Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
         => ( aa(A,B,F3,Y3) = aa(A,B,F3,Z3) ) )
     => equiv_congruent(A,B,R3,F3) ) ).

% congruentI
tff(fact_6579_congruent2D,axiom,
    ! [A: $tType,C: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C)),Y1: A,Z1: A,Y2: B,Z22: B] :
      ( equiv_congruent2(A,B,C,R12,R23,F3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1)),R12))
       => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z22)),R23))
         => ( aa(B,C,aa(A,fun(B,C),F3,Y1),Y2) = aa(B,C,aa(A,fun(B,C),F3,Z1),Z22) ) ) ) ) ).

% congruent2D
tff(fact_6580_congruent2I_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( ! [Y12: A,Z12: A,Y22: B,Z23: B] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y12),Z12)),R12))
         => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z23)),R23))
           => ( aa(B,C,aa(A,fun(B,C),F3,Y12),Y22) = aa(B,C,aa(A,fun(B,C),F3,Z12),Z23) ) ) )
     => equiv_congruent2(A,B,C,R12,R23,F3) ) ).

% congruent2I'
tff(fact_6581_congruent2I,axiom,
    ! [C: $tType,B: $tType,A: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( ! [Y3: A,Z3: A,W: B] :
              ( pp(aa(set(B),bool,member(B,W),A24))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R12))
               => ( aa(B,C,aa(A,fun(B,C),F3,Y3),W) = aa(B,C,aa(A,fun(B,C),F3,Z3),W) ) ) )
         => ( ! [Y3: B,Z3: B,W: A] :
                ( pp(aa(set(A),bool,member(A,W),A15))
               => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y3),Z3)),R23))
                 => ( aa(B,C,aa(A,fun(B,C),F3,W),Y3) = aa(B,C,aa(A,fun(B,C),F3,W),Z3) ) ) )
           => equiv_congruent2(A,B,C,R12,R23,F3) ) ) ) ) ).

% congruent2I
tff(fact_6582_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(A,A)),F3: fun(A,fun(A,B))] :
      ( equiv_equiv(A,A6,R3)
     => ( ! [Y3: A,Z3: A] :
            ( pp(aa(set(A),bool,member(A,Y3),A6))
           => ( pp(aa(set(A),bool,member(A,Z3),A6))
             => ( aa(A,B,aa(A,fun(A,B),F3,Y3),Z3) = aa(A,B,aa(A,fun(A,B),F3,Z3),Y3) ) ) )
       => ( ! [Y3: A,Z3: A,W: A] :
              ( pp(aa(set(A),bool,member(A,W),A6))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
               => ( aa(A,B,aa(A,fun(A,B),F3,W),Y3) = aa(A,B,aa(A,fun(A,B),F3,W),Z3) ) ) )
         => equiv_congruent2(A,A,B,R3,R3,F3) ) ) ) ).

% congruent2_commuteI
tff(fact_6583_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),A3: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F3)
         => ( pp(aa(set(B),bool,member(B,A3),A24))
           => equiv_congruent(A,set(C),R12,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_ajz(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F3),A3)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_6584_UN__equiv__class__type,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(A,A)),F3: fun(A,set(B)),X6: set(A),B5: set(set(B))] :
      ( equiv_equiv(A,A6,R3)
     => ( equiv_congruent(A,set(B),R3,F3)
       => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A6))
               => pp(aa(set(set(B)),bool,member(set(B),aa(A,set(B),F3,X3)),B5)) )
           => pp(aa(set(set(B)),bool,member(set(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X6))),B5)) ) ) ) ) ).

% UN_equiv_class_type
tff(fact_6585_UN__equiv__class__type2,axiom,
    ! [A: $tType,B: $tType,C: $tType,A15: set(A),R12: set(product_prod(A,A)),A24: set(B),R23: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),X13: set(A),X23: set(B),B5: set(set(C))] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A24,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F3)
         => ( pp(aa(set(set(A)),bool,member(set(A),X13),equiv_quotient(A,A15,R12)))
           => ( pp(aa(set(set(B)),bool,member(set(B),X23),equiv_quotient(B,A24,R23)))
             => ( ! [X12: A,X22: B] :
                    ( pp(aa(set(A),bool,member(A,X12),A15))
                   => ( pp(aa(set(B),bool,member(B,X22),A24))
                     => pp(aa(set(set(C)),bool,member(set(C),aa(B,set(C),aa(A,fun(B,set(C)),F3,X12),X22)),B5)) ) )
               => pp(aa(set(set(C)),bool,member(set(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(set(B),fun(A,set(C)),aTP_Lamp_aka(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),F3),X23)),X13))),B5)) ) ) ) ) ) ) ).

% UN_equiv_class_type2
tff(fact_6586_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(A,A)),F3: fun(A,set(B)),X6: set(A),Y6: set(A)] :
      ( equiv_equiv(A,A6,R3)
     => ( equiv_congruent(A,set(B),R3,F3)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X6)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),Y6)) )
         => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
           => ( pp(aa(set(set(A)),bool,member(set(A),Y6),equiv_quotient(A,A6,R3)))
             => ( ! [X3: A,Y3: A] :
                    ( pp(aa(set(A),bool,member(A,X3),A6))
                   => ( pp(aa(set(A),bool,member(A,Y3),A6))
                     => ( ( aa(A,set(B),F3,X3) = aa(A,set(B),F3,Y3) )
                       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3)) ) ) )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_6587_univ__preserves,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(A,A)),F3: fun(A,B),B5: set(B)] :
      ( equiv_equiv(A,A6,R3)
     => ( equiv_congruent(A,B,R3,F3)
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A6))
             => pp(aa(set(B),bool,member(B,aa(A,B,F3,X3)),B5)) )
         => ! [X5: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X5),equiv_quotient(A,A6,R3)))
             => pp(aa(set(B),bool,member(B,bNF_Greatest_univ(A,B,F3,X5)),B5)) ) ) ) ) ).

% univ_preserves
tff(fact_6588_proj__iff,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),A6))
       => ( ( aa(A,set(A),equiv_proj(A,A,R3),X) = aa(A,set(A),equiv_proj(A,A,R3),Y) )
        <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3)) ) ) ) ).

% proj_iff
tff(fact_6589_univ__commute,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(A,A)),F3: fun(A,B),X: A] :
      ( equiv_equiv(A,A6,R3)
     => ( equiv_congruent(A,B,R3,F3)
       => ( pp(aa(set(A),bool,member(A,X),A6))
         => ( bNF_Greatest_univ(A,B,F3,aa(A,set(A),equiv_proj(A,A,R3),X)) = aa(A,B,F3,X) ) ) ) ) ).

% univ_commute
tff(fact_6590_equiv__proj,axiom,
    ! [A: $tType,A6: set(A),R2: set(product_prod(A,A)),Z4: product_prod(A,A)] :
      ( equiv_equiv(A,A6,R2)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),Z4),R2))
       => ( aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),comp(A,set(A),product_prod(A,A),equiv_proj(A,A,R2)),product_fst(A,A)),Z4) = aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),comp(A,set(A),product_prod(A,A),equiv_proj(A,A,R2)),product_snd(A,A)),Z4) ) ) ) ).

% equiv_proj
tff(fact_6591_relImage__proj,axiom,
    ! [A: $tType,A6: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,A6,R2)
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),bNF_Gr4221423524335903396lImage(A,set(A),R2,equiv_proj(A,A,R2))),id_on(set(A),equiv_quotient(A,A6,R2)))) ) ).

% relImage_proj
tff(fact_6592_remove__def,axiom,
    ! [A: $tType,X: A,A6: set(A)] : remove(A,X,A6) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% remove_def
tff(fact_6593_relInvImage__Gr,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),B5: set(A),A6: set(B),F3: fun(B,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))))
     => ( bNF_Gr7122648621184425601vImage(B,A,A6,R2,F3) = relcomp(B,A,B,bNF_Gr(B,A,A6,F3),relcomp(A,A,B,R2,converse(B,A,bNF_Gr(B,A,A6,F3)))) ) ) ).

% relInvImage_Gr
tff(fact_6594_member__remove,axiom,
    ! [A: $tType,X: A,Y: A,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,X),remove(A,Y,A6)))
    <=> ( pp(aa(set(A),bool,member(A,X),A6))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_6595_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R3: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R3)))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R3)) ) ).

% converse_iff
tff(fact_6596_converse__mono,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),S2: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),converse(B,A,R3)),converse(B,A,S2)))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),R3),S2)) ) ).

% converse_mono
tff(fact_6597_pair__set__inverse,axiom,
    ! [A: $tType,B: $tType,P2: fun(B,fun(A,bool))] : converse(B,A,aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P2))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P2))) ).

% pair_set_inverse
tff(fact_6598_below__Id__inv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),converse(A,A,R2)),id2(A)))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),id2(A))) ) ).

% below_Id_inv
tff(fact_6599_finite__wf__eq__wf__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( wf(A,converse(A,A,R2))
      <=> wf(A,R2) ) ) ).

% finite_wf_eq_wf_converse
tff(fact_6600_converse__Times,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B5: set(A)] : converse(B,A,product_Sigma(B,A,A6,aTP_Lamp_ws(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),A6)) ).

% converse_Times
tff(fact_6601_converse__Un,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),S2: set(product_prod(B,A))] : converse(B,A,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))),R3),S2)) = 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))),converse(B,A,R3)),converse(B,A,S2)) ).

% converse_Un
tff(fact_6602_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),converse(A,B,R3))) ) ).

% converseI
tff(fact_6603_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R3: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Yx),converse(B,A,R3)))
     => ~ ! [X3: B,Y3: A] :
            ( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),X3) )
           => ~ pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y3)),R3)) ) ) ).

% converseE
tff(fact_6604_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R3: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R3)))
     => pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R3)) ) ).

% converseD
tff(fact_6605_converse_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: B,A22: A,R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22)),converse(A,B,R3)))
    <=> ? [A9: A,B7: B] :
          ( ( A1 = B7 )
          & ( A22 = A9 )
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B7)),R3)) ) ) ).

% converse.simps
tff(fact_6606_converse_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: B,A22: A,R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22)),converse(A,B,R3)))
     => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A22),A1)),R3)) ) ).

% converse.cases
tff(fact_6607_trancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R3))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R3)))) ) ).

% trancl_converseD
tff(fact_6608_trancl__converseI,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R3))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R3)))) ) ).

% trancl_converseI
tff(fact_6609_rtrancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R3))))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R3))) ) ).

% rtrancl_converseD
tff(fact_6610_rtrancl__converseI,axiom,
    ! [A: $tType,Y: A,X: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R3)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R3)))) ) ).

% rtrancl_converseI
tff(fact_6611_converse__unfold,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A))] : converse(B,A,R3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_akb(set(product_prod(B,A)),fun(A,fun(B,bool)),R3))) ).

% converse_unfold
tff(fact_6612_converse__subset__swap,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),converse(B,A,S2)))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),converse(A,B,R3)),S2)) ) ).

% converse_subset_swap
tff(fact_6613_converse__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R3),S))) = 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))),image2(C,set(product_prod(A,B)),aTP_Lamp_akc(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R3)),S)) ).

% converse_UNION
tff(fact_6614_converse__INTER,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Inf_Inf(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R3),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_akc(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R3)),S)) ).

% converse_INTER
tff(fact_6615_Image__subset__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(B,A)),A6: set(B),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R3),A6)),B5))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,converse(B,A,R3)),aa(set(A),set(A),uminus_uminus(set(A)),B5))))) ) ).

% Image_subset_eq
tff(fact_6616_image2__Gr,axiom,
    ! [A: $tType,B: $tType,C: $tType,A6: set(C),F3: fun(C,A),G3: fun(C,B)] : bNF_Greatest_image2(C,A,B,A6,F3,G3) = relcomp(A,C,B,converse(C,A,bNF_Gr(C,A,A6,F3)),bNF_Gr(C,B,A6,G3)) ).

% image2_Gr
tff(fact_6617_irrefl__tranclI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A] :
      ( ( 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))),inf_inf(set(product_prod(A,A))),converse(A,A,R3)),transitive_rtrancl(A,R3)) = bot_bot(set(product_prod(A,A))) )
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),transitive_trancl(A,R3))) ) ).

% irrefl_tranclI
tff(fact_6618_relImage__Gr,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
     => ( bNF_Gr4221423524335903396lImage(A,B,R2,F3) = relcomp(B,A,B,converse(A,B,bNF_Gr(A,B,A6,F3)),relcomp(A,A,B,R2,bNF_Gr(A,B,A6,F3))) ) ) ).

% relImage_Gr
tff(fact_6619_trans__wf__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
     => ( wf(A,R3)
      <=> ! [A9: A] : wf(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(A),set(A),image(A,A,converse(A,A,R3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A9),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_akd(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),A9)))) ) ) ).

% trans_wf_iff
tff(fact_6620_Image__INT__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(B,A)),A6: set(C),B5: fun(C,set(B))] :
      ( single_valued(A,B,converse(B,A,R3))
     => ( ( A6 != bot_bot(set(C)) )
       => ( aa(set(B),set(A),image(B,A,R3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B5),A6))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_aio(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R3),B5)),A6)) ) ) ) ).

% Image_INT_eq
tff(fact_6621_trans__O__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,R3)),R3)) ) ).

% trans_O_subset
tff(fact_6622_single__valued__subset,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S2))
     => ( single_valued(A,B,S2)
       => single_valued(A,B,R3) ) ) ).

% single_valued_subset
tff(fact_6623_trans__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
    <=> ! [X4: A,Y4: A,Z2: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2)),R3))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z2)),R3)) ) ) ) ).

% trans_def
tff(fact_6624_transI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A,Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3)),R3))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3)),R3)) ) )
     => trans(A,R3) ) ).

% transI
tff(fact_6625_transE,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z4: A] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),R3))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),R3)) ) ) ) ).

% transE
tff(fact_6626_transD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z4: A] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),R3))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),R3)) ) ) ) ).

% transD
tff(fact_6627_single__valuedD,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B)),X: A,Y: B,Z4: B] :
      ( single_valued(A,B,R3)
     => ( pp(aa(set(product_prod(A,B)),bool,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)),R3))
       => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z4)),R3))
         => ( Y = Z4 ) ) ) ) ).

% single_valuedD
tff(fact_6628_single__valuedI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] :
      ( ! [X3: A,Y3: B,Z3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R3))
         => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3)),R3))
           => ( Y3 = Z3 ) ) )
     => single_valued(A,B,R3) ) ).

% single_valuedI
tff(fact_6629_single__valued__def,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,B))] :
      ( single_valued(A,B,R3)
    <=> ! [X4: A,Y4: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4)),R3))
         => ! [Z2: B] :
              ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z2)),R3))
             => ( Y4 = Z2 ) ) ) ) ).

% single_valued_def
tff(fact_6630_trans__reflclI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
     => trans(A,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))),R3),id2(A))) ) ).

% trans_reflclI
tff(fact_6631_single__valued__inter2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( single_valued(A,B,R2)
     => single_valued(A,B,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))),S),R2)) ) ).

% single_valued_inter2
tff(fact_6632_single__valued__inter1,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( single_valued(A,B,R2)
     => single_valued(A,B,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))),R2),S)) ) ).

% single_valued_inter1
tff(fact_6633_trans__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( trans(A,R3)
     => trans(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% trans_Restr
tff(fact_6634_single__valued__confluent,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z4: A] :
      ( single_valued(A,A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3)))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),transitive_rtrancl(A,R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),transitive_rtrancl(A,R3)))
            | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Y)),transitive_rtrancl(A,R3))) ) ) ) ) ).

% single_valued_confluent
tff(fact_6635_single__valued__below__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),id2(A)))
     => single_valued(A,A,R2) ) ).

% single_valued_below_Id
tff(fact_6636_trans__singleton,axiom,
    ! [A: $tType,A3: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),A3),A3)),bot_bot(set(product_prod(A,A))))) ).

% trans_singleton
tff(fact_6637_trans__rtrancl__eq__reflcl,axiom,
    ! [A: $tType,A6: set(product_prod(A,A))] :
      ( trans(A,A6)
     => ( transitive_rtrancl(A,A6) = 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))),A6),id2(A)) ) ) ).

% trans_rtrancl_eq_reflcl
tff(fact_6638_bijective__alt,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( bijective(A,B,R2)
    <=> ( single_valued(A,B,R2)
        & single_valued(B,A,converse(A,B,R2)) ) ) ).

% bijective_alt
tff(fact_6639_wf__finite__segments,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( irrefl(A,R3)
     => ( trans(A,R3)
       => ( ! [X3: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_ake(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),X3))))
         => wf(A,R3) ) ) ) ).

% wf_finite_segments
tff(fact_6640_relation__of__def,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),A6: set(A)] : order_relation_of(A,P2,A6) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_akf(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),P2),A6))) ).

% relation_of_def
tff(fact_6641_irrefl__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( irrefl(A,R3)
    <=> ! [A9: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),A9)),R3)) ) ).

% irrefl_def
tff(fact_6642_irreflI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [A5: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),R2))
     => irrefl(A,R2) ) ).

% irreflI
tff(fact_6643_size__diff__se,axiom,
    ! [A: $tType,T5: A,S: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,T5),aa(multiset(A),set(A),set_mset(A),S)))
     => ( aa(multiset(A),nat,size_size(multiset(A)),S) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T5),zero_zero(multiset(A)))))),one_one(nat)) ) ) ).

% size_diff_se
tff(fact_6644_stable__sort__key__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Sk: fun(fun(B,A),fun(list(B),list(B)))] :
          ( linord3483353639454293061rt_key(B,A,Sk)
        <=> ! [F11: fun(B,A),Xs3: list(B),K3: A] : filter2(B,aa(A,fun(B,bool),aTP_Lamp_qv(fun(B,A),fun(A,fun(B,bool)),F11),K3),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),Sk,F11),Xs3)) = filter2(B,aa(A,fun(B,bool),aTP_Lamp_qv(fun(B,A),fun(A,fun(B,bool)),F11),K3),Xs3) ) ) ).

% stable_sort_key_def
tff(fact_6645_set__mset__union,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(multiset(A),set(A),set_mset(A),M6)),aa(multiset(A),set(A),set_mset(A),N7)) ).

% set_mset_union
tff(fact_6646_mset__diff__cancel1elem,axiom,
    ! [A: $tType,A3: A,B5: multiset(A)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B5)))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),B5) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ).

% mset_diff_cancel1elem
tff(fact_6647_set__mset__Union__mset,axiom,
    ! [A: $tType,MM: multiset(multiset(A))] : aa(multiset(A),set(A),set_mset(A),comm_m7189776963980413722m_mset(multiset(A),MM)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),aa(multiset(multiset(A)),set(multiset(A)),set_mset(multiset(A)),MM))) ).

% set_mset_Union_mset
tff(fact_6648_set__mset__Inf,axiom,
    ! [A: $tType,A6: set(multiset(A))] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A6)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),A6)) ) ) ).

% set_mset_Inf
tff(fact_6649_set__mset__mono,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(multiset(A),set(A),set_mset(A),A6)),aa(multiset(A),set(A),set_mset(A),B5))) ) ).

% set_mset_mono
tff(fact_6650_image__mset__cong__pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,M6: multiset(product_prod(A,B)),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C))] :
      ( ! [X3: A,Y3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),M6)))
         => ( aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(B,C,aa(A,fun(B,C),G3,X3),Y3) ) )
     => ( aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)),M6) = aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3)),M6) ) ) ).

% image_mset_cong_pair
tff(fact_6651_image__mset__cong,axiom,
    ! [B: $tType,A: $tType,M6: multiset(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(multiset(A),set(A),set_mset(A),M6)))
         => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) )
     => ( aa(multiset(A),multiset(B),image_mset(A,B,F3),M6) = aa(multiset(A),multiset(B),image_mset(A,B,G3),M6) ) ) ).

% image_mset_cong
tff(fact_6652_in__image__mset,axiom,
    ! [A: $tType,B: $tType,Y: A,F3: fun(B,A),M6: multiset(B)] :
      ( pp(aa(set(A),bool,member(A,Y),aa(multiset(A),set(A),set_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),M6))))
    <=> pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),image2(B,A,F3),aa(multiset(B),set(B),set_mset(B),M6)))) ) ).

% in_image_mset
tff(fact_6653_ex__Melem__conv,axiom,
    ! [A: $tType,A6: multiset(A)] :
      ( ? [X4: A] : pp(aa(set(A),bool,member(A,X4),aa(multiset(A),set(A),set_mset(A),A6)))
    <=> ( A6 != zero_zero(multiset(A)) ) ) ).

% ex_Melem_conv
tff(fact_6654_union__iff,axiom,
    ! [A: $tType,A3: A,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5))))
    <=> ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A6)))
        | pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B5))) ) ) ).

% union_iff
tff(fact_6655_mset__right__cancel__union,axiom,
    ! [A: $tType,A3: A,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5))))
     => ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B5)))
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A6))) ) ) ).

% mset_right_cancel_union
tff(fact_6656_mset__left__cancel__union,axiom,
    ! [A: $tType,A3: A,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5))))
     => ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A6)))
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B5))) ) ) ).

% mset_left_cancel_union
tff(fact_6657_mset__un__cases,axiom,
    ! [A: $tType,A3: A,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5))))
     => ( ~ pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A6)))
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B5))) ) ) ).

% mset_un_cases
tff(fact_6658_multiset__induct__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(multiset(A),bool),M6: multiset(A)] :
          ( pp(aa(multiset(A),bool,P2,zero_zero(multiset(A))))
         => ( ! [X3: A,M11: multiset(A)] :
                ( pp(aa(multiset(A),bool,P2,M11))
               => ( ! [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),aa(multiset(A),set(A),set_mset(A),M11)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X3)) )
                 => pp(aa(multiset(A),bool,P2,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X3),M11))) ) )
           => pp(aa(multiset(A),bool,P2,M6)) ) ) ) ).

% multiset_induct_max
tff(fact_6659_multiset__induct__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(multiset(A),bool),M6: multiset(A)] :
          ( pp(aa(multiset(A),bool,P2,zero_zero(multiset(A))))
         => ( ! [X3: A,M11: multiset(A)] :
                ( pp(aa(multiset(A),bool,P2,M11))
               => ( ! [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),aa(multiset(A),set(A),set_mset(A),M11)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa)) )
                 => pp(aa(multiset(A),bool,P2,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X3),M11))) ) )
           => pp(aa(multiset(A),bool,P2,M6)) ) ) ) ).

% multiset_induct_min
tff(fact_6660_mset__left__cancel__elem,axiom,
    ! [A: $tType,A3: A,B2: A,A6: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))),A6))))
     => ( ( A3 != B2 )
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A6))) ) ) ).

% mset_left_cancel_elem
tff(fact_6661_mset__right__cancel__elem,axiom,
    ! [A: $tType,A3: A,A6: multiset(A),B2: A] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))))))
     => ( ( A3 != B2 )
       => pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A6))) ) ) ).

% mset_right_cancel_elem
tff(fact_6662_multi__member__this,axiom,
    ! [A: $tType,X: A,XS: multiset(A)] : pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))),XS)))) ).

% multi_member_this
tff(fact_6663_multi__member__skip,axiom,
    ! [A: $tType,X: A,XS: multiset(A),Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),XS)))
     => pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Y),zero_zero(multiset(A)))),XS)))) ) ).

% multi_member_skip
tff(fact_6664_sum__mset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: multiset(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),aa(multiset(B),set(B),set_mset(B),K5)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,F3),K5))),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),K5)))) ) ) ).

% sum_mset_mono
tff(fact_6665_mset__2dist2__cases,axiom,
    ! [A: $tType,A3: A,B2: A,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)))
     => ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),A6))
       => ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),B5))
         => ( ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),A6)))
             => ~ pp(aa(set(A),bool,member(A,B2),aa(multiset(A),set(A),set_mset(A),B5))) )
           => ~ ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B5)))
               => ~ pp(aa(set(A),bool,member(A,B2),aa(multiset(A),set(A),set_mset(A),A6))) ) ) ) ) ) ).

% mset_2dist2_cases
tff(fact_6666_mset__union__subset__s,axiom,
    ! [A: $tType,A3: A,B5: multiset(A),C4: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),B5)),C4))
     => ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),C4)))
        & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),C4)) ) ) ).

% mset_union_subset_s
tff(fact_6667_mset__le__mono__add__single,axiom,
    ! [A: $tType,A3: A,Ys2: multiset(A),B2: A,Ws2: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),Ys2)))
     => ( pp(aa(set(A),bool,member(A,B2),aa(multiset(A),set(A),set_mset(A),Ws2)))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ys2),Ws2))) ) ) ).

% mset_le_mono_add_single
tff(fact_6668_nth__mem__mset,axiom,
    ! [A: $tType,I2: nat,Ls2: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ls2)))
     => pp(aa(set(A),bool,member(A,aa(nat,A,nth(A,Ls2),I2)),aa(multiset(A),set(A),set_mset(A),mset(A,Ls2)))) ) ).

% nth_mem_mset
tff(fact_6669_insert__DiffM2,axiom,
    ! [A: $tType,X: A,M6: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6)))
     => ( 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)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))) = M6 ) ) ).

% insert_DiffM2
tff(fact_6670_diff__union__single__conv,axiom,
    ! [A: $tType,A3: A,J4: multiset(A),I5: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),J4)))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),I5),J4)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),J4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))) ) ) ).

% diff_union_single_conv
tff(fact_6671_mset__un__single__un__cases,axiom,
    ! [A: $tType,A3: A,A6: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4) )
     => ( ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),B5)))
         => ( A6 != 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)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))),C4) ) )
       => ~ ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),C4)))
           => ( A6 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))) ) ) ) ) ).

% mset_un_single_un_cases
tff(fact_6672_diff__union__single__conv2,axiom,
    ! [A: $tType,A3: A,J4: multiset(A),I5: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),J4)))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),J4),I5)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(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)),minus_minus(multiset(A)),J4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))),I5) ) ) ).

% diff_union_single_conv2
tff(fact_6673_mset__union__diff__comm,axiom,
    ! [A: $tType,T5: A,S: multiset(A),T8: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,T5),aa(multiset(A),set(A),set_mset(A),S)))
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T8),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T5),zero_zero(multiset(A))))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),T8),S)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T5),zero_zero(multiset(A)))) ) ) ).

% mset_union_diff_comm
tff(fact_6674_mset__contains__eq,axiom,
    ! [A: $tType,M: A,M6: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,M),aa(multiset(A),set(A),set_mset(A),M6)))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M),zero_zero(multiset(A))))) = M6 ) ) ).

% mset_contains_eq
tff(fact_6675_mset__size1elem,axiom,
    ! [A: $tType,P2: multiset(A),Q5: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P2)),one_one(nat)))
     => ( pp(aa(set(A),bool,member(A,Q5),aa(multiset(A),set(A),set_mset(A),P2)))
       => ( P2 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q5),zero_zero(multiset(A))) ) ) ) ).

% mset_size1elem
tff(fact_6676_size__Diff2__less,axiom,
    ! [A: $tType,X: A,M6: multiset(A),Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6)))
     => ( pp(aa(set(A),bool,member(A,Y),aa(multiset(A),set(A),set_mset(A),M6)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Y),zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ) ) ).

% size_Diff2_less
tff(fact_6677_size__Diff1__less,axiom,
    ! [A: $tType,X: A,M6: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ) ).

% size_Diff1_less
tff(fact_6678_sum__mset_Oremove,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [X: A,A6: multiset(A)] :
          ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),A6)))
         => ( comm_m7189776963980413722m_mset(A,A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),comm_m7189776963980413722m_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X),zero_zero(multiset(A)))))) ) ) ) ).

% sum_mset.remove
tff(fact_6679_subset__mset_Osum__mset__mono,axiom,
    ! [A: $tType,B: $tType,K5: multiset(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( ! [I3: B] :
          ( pp(aa(set(B),bool,member(B,I3),aa(multiset(B),set(B),set_mset(B),K5)))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,I3)),aa(B,multiset(A),G3,I3))) )
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),comm_monoid_sum_mset(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),aa(multiset(B),multiset(multiset(A)),image_mset(B,multiset(A),F3),K5))),comm_monoid_sum_mset(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),aa(multiset(B),multiset(multiset(A)),image_mset(B,multiset(A),G3),K5)))) ) ).

% subset_mset.sum_mset_mono
tff(fact_6680_subset__mset_Osum__mset__0__iff,axiom,
    ! [A: $tType,M6: multiset(multiset(A))] :
      ( ( comm_monoid_sum_mset(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),M6) = zero_zero(multiset(A)) )
    <=> ! [X4: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),aa(multiset(multiset(A)),set(multiset(A)),set_mset(multiset(A)),M6)))
         => ( X4 = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.sum_mset_0_iff
tff(fact_6681_mult__implies__one__step,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),M6: multiset(A),N7: multiset(A)] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M6),N7)),mult(A,R3)))
       => ? [I7: multiset(A),J6: multiset(A)] :
            ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),J6) )
            & ? [K8: multiset(A)] :
                ( ( M6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),K8) )
                & ( J6 != zero_zero(multiset(A)) )
                & ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),aa(multiset(A),set(A),set_mset(A),K8)))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),aa(multiset(A),set(A),set_mset(A),J6)))
                        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa4)),R3)) ) ) ) ) ) ) ).

% mult_implies_one_step
tff(fact_6682_multiset_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: multiset(A),B2: multiset(B)] :
      ( pp(aa(multiset(B),bool,aa(multiset(A),fun(multiset(B),bool),rel_mset(A,B,R2),A3),B2))
    <=> ? [Z2: multiset(product_prod(A,B))] :
          ( pp(aa(set(multiset(product_prod(A,B))),bool,member(multiset(product_prod(A,B)),Z2),aa(fun(multiset(product_prod(A,B)),bool),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_akg(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),R2))))
          & ( aa(multiset(product_prod(A,B)),multiset(A),image_mset(product_prod(A,B),A,product_fst(A,B)),Z2) = A3 )
          & ( aa(multiset(product_prod(A,B)),multiset(B),image_mset(product_prod(A,B),B,product_snd(A,B)),Z2) = B2 ) ) ) ).

% multiset.in_rel
tff(fact_6683_multiset_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: multiset(A),Y: multiset(B)] :
      ( pp(aa(multiset(B),bool,aa(multiset(C),fun(multiset(B),bool),rel_mset(C,B,Sb),aa(multiset(A),multiset(C),image_mset(A,C,I2),X)),Y))
    <=> pp(aa(multiset(B),bool,aa(multiset(A),fun(multiset(B),bool),rel_mset(A,B,aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_abg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% multiset.rel_map(1)
tff(fact_6684_multiset_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: multiset(A),G3: fun(B,C),Y: multiset(B)] :
      ( pp(aa(multiset(C),bool,aa(multiset(A),fun(multiset(C),bool),rel_mset(A,C,Sa),X),aa(multiset(B),multiset(C),image_mset(B,C,G3),Y)))
    <=> pp(aa(multiset(B),bool,aa(multiset(A),fun(multiset(B),bool),rel_mset(A,B,aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_abf(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G3)),X),Y)) ) ).

% multiset.rel_map(2)
tff(fact_6685_Multiset_Omono__mult,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R6))
     => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),bool),ord_less_eq(set(product_prod(multiset(A),multiset(A)))),mult(A,R3)),mult(A,R6))) ) ).

% Multiset.mono_mult
tff(fact_6686_multiset_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra2))
     => pp(aa(fun(multiset(A),fun(multiset(B),bool)),bool,aa(fun(multiset(A),fun(multiset(B),bool)),fun(fun(multiset(A),fun(multiset(B),bool)),bool),ord_less_eq(fun(multiset(A),fun(multiset(B),bool))),rel_mset(A,B,R2)),rel_mset(A,B,Ra2))) ) ).

% multiset.rel_mono
tff(fact_6687_mult__cancel,axiom,
    ! [A: $tType,S2: set(product_prod(A,A)),X6: multiset(A),Z5: multiset(A),Y6: multiset(A)] :
      ( trans(A,S2)
     => ( irrefl(A,S2)
       => ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X6),Z5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y6),Z5))),mult(A,S2)))
        <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X6),Y6)),mult(A,S2))) ) ) ) ).

% mult_cancel
tff(fact_6688_one__step__implies__mult,axiom,
    ! [A: $tType,J4: multiset(A),K5: multiset(A),R3: set(product_prod(A,A)),I5: multiset(A)] :
      ( ( J4 != zero_zero(multiset(A)) )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(multiset(A),set(A),set_mset(A),K5)))
           => ? [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),aa(multiset(A),set(A),set_mset(A),J4)))
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa)),R3)) ) )
       => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),K5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),J4))),mult(A,R3))) ) ) ).

% one_step_implies_mult
tff(fact_6689_multp__code__iff__mult,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P2: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( irrefl(A,R2)
     => ( trans(A,R2)
       => ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Y3))
            <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R2)) )
         => ( multp_code(A,P2,N7,M6)
          <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N7),M6)),mult(A,R2))) ) ) ) ) ).

% multp_code_iff_mult
tff(fact_6690_multeqp__code__iff__reflcl__mult,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P2: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( irrefl(A,R2)
     => ( trans(A,R2)
       => ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Y3))
            <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R2)) )
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multeqp_code(A,P2),N7),M6))
          <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N7),M6)),aa(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A))),aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A)))),sup_sup(set(product_prod(multiset(A),multiset(A)))),mult(A,R2)),id2(multiset(A))))) ) ) ) ) ).

% multeqp_code_iff_reflcl_mult
tff(fact_6691_mult1E,axiom,
    ! [A: $tType,N7: multiset(A),M6: multiset(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N7),M6)),mult1(A,R3)))
     => ~ ! [A5: A,M0: multiset(A)] :
            ( ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A5),M0) )
           => ! [K8: multiset(A)] :
                ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K8) )
               => ~ ! [B10: A] :
                      ( pp(aa(set(A),bool,member(A,B10),aa(multiset(A),set(A),set_mset(A),K8)))
                     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B10),A5)),R3)) ) ) ) ) ).

% mult1E
tff(fact_6692_mult1I,axiom,
    ! [A: $tType,M6: multiset(A),A3: A,M02: multiset(A),N7: multiset(A),K5: multiset(A),R3: set(product_prod(A,A))] :
      ( ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M02) )
     => ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K5) )
       => ( ! [B4: A] :
              ( pp(aa(set(A),bool,member(A,B4),aa(multiset(A),set(A),set_mset(A),K5)))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A3)),R3)) )
         => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N7),M6)),mult1(A,R3))) ) ) ) ).

% mult1I
tff(fact_6693_mult1__union,axiom,
    ! [A: $tType,B5: multiset(A),D5: multiset(A),R3: set(product_prod(A,A)),C4: multiset(A)] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),B5),D5)),mult1(A,R3)))
     => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D5))),mult1(A,R3))) ) ).

% mult1_union
tff(fact_6694_mono__mult1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R6))
     => pp(aa(set(product_prod(multiset(A),multiset(A))),bool,aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),bool),ord_less_eq(set(product_prod(multiset(A),multiset(A)))),mult1(A,R3)),mult1(A,R6))) ) ).

% mono_mult1
tff(fact_6695_less__add,axiom,
    ! [A: $tType,N7: multiset(A),A3: A,M02: multiset(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M02))),mult1(A,R3)))
     => ( ? [M11: multiset(A)] :
            ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M11),M02)),mult1(A,R3)))
            & ( N7 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M11) ) )
        | ? [K8: multiset(A)] :
            ( ! [B10: A] :
                ( pp(aa(set(A),bool,member(A,B10),aa(multiset(A),set(A),set_mset(A),K8)))
               => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B10),A3)),R3)) )
            & ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K8) ) ) ) ) ).

% less_add
tff(fact_6696_mult1__lessE,axiom,
    ! [A: $tType,N7: multiset(A),M6: multiset(A),R3: fun(A,fun(A,bool))] :
      ( pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N7),M6)),mult1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3)))))
     => ( asymp(A,R3)
       => ~ ! [A5: A,M0: multiset(A)] :
              ( ( M6 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A5),M0) )
             => ! [K8: multiset(A)] :
                  ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K8) )
                 => ( ~ pp(aa(set(A),bool,member(A,A5),aa(multiset(A),set(A),set_mset(A),K8)))
                   => ~ ! [B10: A] :
                          ( pp(aa(set(A),bool,member(A,B10),aa(multiset(A),set(A),set_mset(A),K8)))
                         => pp(aa(A,bool,aa(A,fun(A,bool),R3,B10),A5)) ) ) ) ) ) ) ).

% mult1_lessE
tff(fact_6697_count__image__mset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A6: multiset(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),A6)),X) = groups7311177749621191930dd_sum(B,nat,aa(multiset(B),fun(B,nat),count(B),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),aa(multiset(B),set(B),set_mset(B),A6))) ).

% count_image_mset
tff(fact_6698_mset__empty__count,axiom,
    ! [A: $tType,M6: multiset(A)] :
      ( ! [P6: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),P6) = zero_zero(nat)
    <=> ( M6 = zero_zero(multiset(A)) ) ) ).

% mset_empty_count
tff(fact_6699_count__union,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A),A3: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),A3)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N7),A3)) ).

% count_union
tff(fact_6700_count__greater__zero__iff,axiom,
    ! [A: $tType,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6))) ) ).

% count_greater_zero_iff
tff(fact_6701_count__greater__eq__one__iff,axiom,
    ! [A: $tType,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6))) ) ).

% count_greater_eq_one_iff
tff(fact_6702_count__greater__eq__Suc__zero__iff,axiom,
    ! [A: $tType,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6))) ) ).

% count_greater_eq_Suc_zero_iff
tff(fact_6703_asymp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,aTP_Lamp_ax(A,fun(A,bool))) ) ).

% asymp_greater
tff(fact_6704_asymp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,ord_less(A)) ) ).

% asymp_less
tff(fact_6705_plus__multiset_Orep__eq,axiom,
    ! [A: $tType,X: multiset(A),Xa2: multiset(A),X5: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),Xa2)),X5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),X),X5)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Xa2),X5)) ).

% plus_multiset.rep_eq
tff(fact_6706_count__sum,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),A6: set(B),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),groups7311177749621191930dd_sum(B,multiset(A),F3,A6)),X) = groups7311177749621191930dd_sum(B,nat,aa(A,fun(B,nat),aTP_Lamp_akh(fun(B,multiset(A)),fun(A,fun(B,nat)),F3),X),A6) ).

% count_sum
tff(fact_6707_mset__subset__eqI,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] :
      ( ! [A5: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),A6),A5)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),B5),A5)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5)) ) ).

% mset_subset_eqI
tff(fact_6708_subseteq__mset__def,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5))
    <=> ! [A9: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),A6),A9)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),B5),A9))) ) ).

% subseteq_mset_def
tff(fact_6709_mset__subset__eq__count,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),A3: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),B5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),A6),A3)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),B5),A3))) ) ).

% mset_subset_eq_count
tff(fact_6710_in__diff__count,axiom,
    ! [A: $tType,A3: A,M6: multiset(A),N7: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,A3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),N7))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N7),A3)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),A3))) ) ).

% in_diff_count
tff(fact_6711_count__ne__remove,axiom,
    ! [A: $tType,X: A,T5: A,S: multiset(A)] :
      ( ( X != T5 )
     => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),S),X) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T5),zero_zero(multiset(A))))),X) ) ) ).

% count_ne_remove
tff(fact_6712_count__in__diffI,axiom,
    ! [A: $tType,N7: multiset(A),X: A,M6: multiset(A)] :
      ( ! [N4: nat] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N7),X) != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X))
     => pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),N7)))) ) ).

% count_in_diffI
tff(fact_6713_set__mset__def,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),set(A),set_mset(A),M6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aki(multiset(A),fun(A,bool),M6)) ).

% set_mset_def
tff(fact_6714_set__mset__diff,axiom,
    ! [A: $tType,M6: multiset(A),N7: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),N7)) = aa(fun(A,bool),set(A),collect(A),aa(multiset(A),fun(A,bool),aTP_Lamp_akj(multiset(A),fun(multiset(A),fun(A,bool)),M6),N7)) ).

% set_mset_diff
tff(fact_6715_count__mset,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset(A,Xs)),X) = aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs)) ).

% count_mset
tff(fact_6716_count__image__mset_H,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),X6: multiset(B),Y: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(B),multiset(A),image_mset(B,A,F3),X6)),Y) = groups7311177749621191930dd_sum(B,nat,aa(multiset(B),fun(B,nat),count(B),X6),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aa(multiset(B),fun(A,fun(B,bool)),aTP_Lamp_akk(fun(B,A),fun(multiset(B),fun(A,fun(B,bool))),F3),X6),Y))) ).

% count_image_mset'
tff(fact_6717_size__multiset__overloaded__eq,axiom,
    ! [B: $tType,X: multiset(B)] : aa(multiset(B),nat,size_size(multiset(B)),X) = groups7311177749621191930dd_sum(B,nat,aa(multiset(B),fun(B,nat),count(B),X),aa(multiset(B),set(B),set_mset(B),X)) ).

% size_multiset_overloaded_eq
tff(fact_6718_count,axiom,
    ! [A: $tType,X: multiset(A)] : pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),aa(multiset(A),fun(A,nat),count(A),X)),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool)))) ).

% count
tff(fact_6719_count__cases,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool))))
     => ~ ! [X3: multiset(A)] : Y != aa(multiset(A),fun(A,nat),count(A),X3) ) ).

% count_cases
tff(fact_6720_count__induct,axiom,
    ! [A: $tType,Y: fun(A,nat),P2: fun(fun(A,nat),bool)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool))))
     => ( ! [X3: multiset(A)] : pp(aa(fun(A,nat),bool,P2,aa(multiset(A),fun(A,nat),count(A),X3)))
       => pp(aa(fun(A,nat),bool,P2,Y)) ) ) ).

% count_induct
tff(fact_6721_Inf__multiset_Orep__eq,axiom,
    ! [A: $tType,X: set(multiset(A)),X5: A] :
      ( ( ( aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X) = bot_bot(set(fun(A,nat))) )
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X)),X5) = zero_zero(nat) ) )
      & ( ( aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X) != bot_bot(set(fun(A,nat))) )
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),X)),X5) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_akm(A,fun(fun(A,nat),nat),X5)),aa(set(multiset(A)),set(fun(A,nat)),image2(multiset(A),fun(A,nat),count(A)),X))) ) ) ) ).

% Inf_multiset.rep_eq
tff(fact_6722_count__Inf__multiset__nonempty,axiom,
    ! [A: $tType,A6: set(multiset(A)),X: A] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A6)),X) = aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_akn(A,fun(multiset(A),nat),X)),A6)) ) ) ).

% count_Inf_multiset_nonempty
tff(fact_6723_count__mset__gt__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset(A,Xs)),X))) ) ).

% count_mset_gt_0
tff(fact_6724_in__diff__countE,axiom,
    ! [A: $tType,X: A,M6: multiset(A),N7: multiset(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),N7))))
     => ~ ! [N4: nat] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X) != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,N4)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N7),X)) ) ).

% in_diff_countE
tff(fact_6725_sum__mset__delta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C3: A,A6: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_ako(B,fun(A,fun(B,A)),Y),C3)),A6)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A6),Y))) ) ).

% sum_mset_delta'
tff(fact_6726_sum__mset__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Y: B,C3: A,A6: multiset(B)] : comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_akp(B,fun(A,fun(B,A)),Y),C3)),A6)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(nat,A,semiring_1_of_nat(A),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A6),Y))) ) ).

% sum_mset_delta
tff(fact_6727_size__multiset__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),M6: multiset(A)] : aa(multiset(A),nat,size_multiset(A,F3),M6) = groups7311177749621191930dd_sum(A,nat,aa(multiset(A),fun(A,nat),aTP_Lamp_akq(fun(A,nat),fun(multiset(A),fun(A,nat)),F3),M6),aa(multiset(A),set(A),set_mset(A),M6)) ).

% size_multiset_eq
tff(fact_6728_bdd__above__multiset__imp__finite__support,axiom,
    ! [A: $tType,A6: set(multiset(A))] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
       => pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),aTP_Lamp_akr(multiset(A),set(A))),A6)))) ) ) ).

% bdd_above_multiset_imp_finite_support
tff(fact_6729_Sup__multiset__in__multiset,axiom,
    ! [A: $tType,A6: set(multiset(A))] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
       => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aks(set(multiset(A)),fun(A,bool),A6)))) ) ) ).

% Sup_multiset_in_multiset
tff(fact_6730_subset__mset_Obdd__above__Un,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A6),B5)))
    <=> ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
        & pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),B5)) ) ) ).

% subset_mset.bdd_above_Un
tff(fact_6731_subset__mset_Obdd__above__UN,axiom,
    ! [A: $tType,B: $tType,I5: set(B),A6: fun(B,set(multiset(A)))] :
      ( pp(aa(set(B),bool,finite_finite2(B),I5))
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(set(multiset(A))),set(multiset(A)),complete_Sup_Sup(set(multiset(A))),aa(set(B),set(set(multiset(A))),image2(B,set(multiset(A)),A6),I5))))
      <=> ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),I5))
           => pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(B,set(multiset(A)),A6,X4))) ) ) ) ).

% subset_mset.bdd_above_UN
tff(fact_6732_subset__mset_Obdd__above__mono,axiom,
    ! [A: $tType,B5: set(multiset(A)),A6: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),B5))
     => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A6),B5))
       => pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6)) ) ) ).

% subset_mset.bdd_above_mono
tff(fact_6733_preorder_Obdd__above_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : condit8047198070973881523_above(A,Less_eq) = condit8047198070973881523_above(A,Less_eq) ).

% preorder.bdd_above.cong
tff(fact_6734_subset__mset_OcSup__subset__mono,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),B5))
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A6),B5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A6)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B5))) ) ) ) ).

% subset_mset.cSup_subset_mono
tff(fact_6735_subset__mset_OcSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType,A6: set(B),G3: fun(B,multiset(A)),B5: set(B),F3: fun(B,multiset(A))] :
      ( ( A6 != bot_bot(set(B)) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B5)))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A6))
               => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F3,X3)),aa(B,multiset(A),G3,X3))) )
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B5)))) ) ) ) ) ).

% subset_mset.cSUP_subset_mono
tff(fact_6736_set__mset__Sup,axiom,
    ! [A: $tType,A6: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
     => ( aa(multiset(A),set(A),set_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(multiset(A)),set(set(A)),image2(multiset(A),set(A),set_mset(A)),A6)) ) ) ).

% set_mset_Sup
tff(fact_6737_count__Sup__multiset__nonempty,axiom,
    ! [A: $tType,A6: set(multiset(A)),X: A] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
       => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A6)),X) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_akn(A,fun(multiset(A),nat),X)),A6)) ) ) ) ).

% count_Sup_multiset_nonempty
tff(fact_6738_subset__mset_Omono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A6: fun(C,multiset(A)),I5: set(C)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A6),I5)))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_akt(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F3),A6)),I5))),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A6),I5))))) ) ) ) ) ).

% subset_mset.mono_cSUP
tff(fact_6739_subset__mset_Omono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A6: set(multiset(A))] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
           => ( ( A6 != bot_bot(set(multiset(A))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F3),A6))),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A6)))) ) ) ) ) ).

% subset_mset.mono_cSup
tff(fact_6740_order_Omono_Ocong,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [Less_eq: fun(A,fun(A,bool))] : mono(A,B,Less_eq) = mono(A,B,Less_eq) ) ).

% order.mono.cong
tff(fact_6741_subset__mset_Omono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
        <=> ! [X4: multiset(A),Y4: multiset(A)] :
              ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X4),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X4)),aa(multiset(A),B,F3,Y4))) ) ) ) ).

% subset_mset.mono_def
tff(fact_6742_subset__mset_OmonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B)] :
          ( ! [X3: multiset(A),Y3: multiset(A)] :
              ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X3)),aa(multiset(A),B,F3,Y3))) )
         => pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3)) ) ) ).

% subset_mset.monoI
tff(fact_6743_subset__mset_OmonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B),X: multiset(A),Y: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X)),aa(multiset(A),B,F3,Y))) ) ) ) ).

% subset_mset.monoE
tff(fact_6744_subset__mset_OmonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(multiset(A),B),X: multiset(A),Y: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,X)),aa(multiset(A),B,F3,Y))) ) ) ) ).

% subset_mset.monoD
tff(fact_6745_Inf__multiset__def,axiom,
    ! [A: $tType] : complete_Inf_Inf(multiset(A)) = aa(fun(set(fun(A,nat)),fun(A,nat)),fun(set(multiset(A)),multiset(A)),map_fun(set(multiset(A)),set(fun(A,nat)),fun(A,nat),multiset(A),image2(multiset(A),fun(A,nat),count(A)),abs_multiset(A)),aTP_Lamp_aku(set(fun(A,nat)),fun(A,nat))) ).

% Inf_multiset_def
tff(fact_6746_Sup__multiset__def,axiom,
    ! [A: $tType,A6: set(multiset(A))] :
      ( ( ( ( A6 != bot_bot(set(multiset(A))) )
          & pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6)) )
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A6) = aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_akv(set(multiset(A)),fun(A,nat),A6)) ) )
      & ( ~ ( ( A6 != bot_bot(set(multiset(A))) )
            & pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6)) )
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A6) = zero_zero(multiset(A)) ) ) ) ).

% Sup_multiset_def
tff(fact_6747_count__Abs__multiset,axiom,
    ! [A: $tType,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_kx(fun(A,nat),fun(A,bool),F3))))
     => ( aa(multiset(A),fun(A,nat),count(A),aa(fun(A,nat),multiset(A),abs_multiset(A),F3)) = F3 ) ) ).

% count_Abs_multiset
tff(fact_6748_zero__multiset__def,axiom,
    ! [A: $tType] : zero_zero(multiset(A)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aTP_Lamp_akw(A,nat)) ).

% zero_multiset_def
tff(fact_6749_Abs__multiset__inject,axiom,
    ! [A: $tType,X: fun(A,nat),Y: fun(A,nat)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),X),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool))))
     => ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool))))
       => ( ( aa(fun(A,nat),multiset(A),abs_multiset(A),X) = aa(fun(A,nat),multiset(A),abs_multiset(A),Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_multiset_inject
tff(fact_6750_Abs__multiset__induct,axiom,
    ! [A: $tType,P2: fun(multiset(A),bool),X: multiset(A)] :
      ( ! [Y3: fun(A,nat)] :
          ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y3),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool))))
         => pp(aa(multiset(A),bool,P2,aa(fun(A,nat),multiset(A),abs_multiset(A),Y3))) )
     => pp(aa(multiset(A),bool,P2,X)) ) ).

% Abs_multiset_induct
tff(fact_6751_Abs__multiset__cases,axiom,
    ! [A: $tType,X: multiset(A)] :
      ~ ! [Y3: fun(A,nat)] :
          ( ( X = aa(fun(A,nat),multiset(A),abs_multiset(A),Y3) )
         => ~ pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y3),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool)))) ) ).

% Abs_multiset_cases
tff(fact_6752_plus__multiset__def,axiom,
    ! [A: $tType] : plus_plus(multiset(A)) = aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(multiset(A),fun(multiset(A),multiset(A))),map_fun(multiset(A),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),count(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_akx(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% plus_multiset_def
tff(fact_6753_minus__multiset__def,axiom,
    ! [A: $tType] : minus_minus(multiset(A)) = aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(multiset(A),fun(multiset(A),multiset(A))),map_fun(multiset(A),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),count(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_aky(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).

% minus_multiset_def
tff(fact_6754_Abs__multiset__inverse,axiom,
    ! [A: $tType,Y: fun(A,nat)] :
      ( pp(aa(set(fun(A,nat)),bool,member(fun(A,nat),Y),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool))))
     => ( aa(multiset(A),fun(A,nat),count(A),aa(fun(A,nat),multiset(A),abs_multiset(A),Y)) = Y ) ) ).

% Abs_multiset_inverse
tff(fact_6755_add__mset__def,axiom,
    ! [A: $tType] : add_mset(A) = aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(A,fun(multiset(A),multiset(A))),map_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_akz(A,fun(fun(A,nat),fun(A,nat)))) ).

% add_mset_def
tff(fact_6756_subset__mset_Omono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A6: fun(C,multiset(A)),I5: set(C)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A6),I5)))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(C),set(multiset(A)),image2(C,multiset(A),A6),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_akt(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),F3),A6)),I5)))) ) ) ) ) ).

% subset_mset.mono_cINF
tff(fact_6757_subset__mset_Omono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(multiset(A),B),A6: set(multiset(A))] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A6))
           => ( ( A6 != bot_bot(set(multiset(A))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A6))),aa(set(B),B,complete_Inf_Inf(B),aa(set(multiset(A)),set(B),image2(multiset(A),B,F3),A6)))) ) ) ) ) ).

% subset_mset.mono_cInf
tff(fact_6758_subset__mset_Obdd__below__Un,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A6),B5)))
    <=> ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A6))
        & pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B5)) ) ) ).

% subset_mset.bdd_below_Un
tff(fact_6759_subset__mset_Obdd__below__UN,axiom,
    ! [A: $tType,B: $tType,I5: set(B),A6: fun(B,set(multiset(A)))] :
      ( pp(aa(set(B),bool,finite_finite2(B),I5))
     => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(set(multiset(A))),set(multiset(A)),complete_Sup_Sup(set(multiset(A))),aa(set(B),set(set(multiset(A))),image2(B,set(multiset(A)),A6),I5))))
      <=> ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),I5))
           => pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(B,set(multiset(A)),A6,X4))) ) ) ) ).

% subset_mset.bdd_below_UN
tff(fact_6760_preorder_Obdd__below_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : condit8119078960628432327_below(A,Less_eq) = condit8119078960628432327_below(A,Less_eq) ).

% preorder.bdd_below.cong
tff(fact_6761_subset__mset_Obdd__below__mono,axiom,
    ! [A: $tType,B5: set(multiset(A)),A6: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B5))
     => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A6),B5))
       => pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A6)) ) ) ).

% subset_mset.bdd_below_mono
tff(fact_6762_subset__mset_OcInf__superset__mono,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B5))
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A6),B5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B5)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A6))) ) ) ) ).

% subset_mset.cInf_superset_mono
tff(fact_6763_subset__mset_OcINF__superset__mono,axiom,
    ! [A: $tType,B: $tType,A6: set(B),G3: fun(B,multiset(A)),B5: set(B),F3: fun(B,multiset(A))] :
      ( ( A6 != bot_bot(set(B)) )
     => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B5)))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),B5))
               => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),G3,X3)),aa(B,multiset(A),F3,X3))) )
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),B5))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))) ) ) ) ) ).

% subset_mset.cINF_superset_mono
tff(fact_6764_repeat__mset__def,axiom,
    ! [A: $tType] : repeat_mset(A) = aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(nat,fun(multiset(A),multiset(A))),map_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(nat),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_ala(nat,fun(fun(A,nat),fun(A,nat)))) ).

% repeat_mset_def
tff(fact_6765_fold__mset__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),S2: B,M6: multiset(A)] : fold_mset(A,B,F3,S2,M6) = finite_fold(A,B,aa(multiset(A),fun(A,fun(B,B)),aTP_Lamp_alb(fun(A,fun(B,B)),fun(multiset(A),fun(A,fun(B,B))),F3),M6),S2,aa(multiset(A),set(A),set_mset(A),M6)) ).

% fold_mset_def
tff(fact_6766_repeat__mset__distrib2,axiom,
    ! [A: $tType,N: nat,A6: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),N),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),N),A6)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),N),B5)) ).

% repeat_mset_distrib2
tff(fact_6767_repeat__mset__Suc,axiom,
    ! [A: $tType,N: nat,M6: multiset(A)] : aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,suc,N)),M6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),N),M6)) ).

% repeat_mset_Suc
tff(fact_6768_in__mset__fold__plus__iff,axiom,
    ! [A: $tType,X: A,M6: multiset(A),NN: multiset(multiset(A))] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),fold_mset(multiset(A),multiset(A),plus_plus(multiset(A)),M6,NN))))
    <=> ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),M6)))
        | ? [N13: multiset(A)] :
            ( pp(aa(set(multiset(A)),bool,member(multiset(A),N13),aa(multiset(multiset(A)),set(multiset(A)),set_mset(multiset(A)),NN)))
            & pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),N13))) ) ) ) ).

% in_mset_fold_plus_iff
tff(fact_6769_repeat__mset__distrib,axiom,
    ! [A: $tType,M: nat,N: nat,A6: multiset(A)] : aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),M),A6)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),N),A6)) ).

% repeat_mset_distrib
tff(fact_6770_left__add__mult__distrib__mset,axiom,
    ! [A: $tType,I2: nat,U: multiset(A),J2: nat,K2: 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(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J2),U)),K2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),U)),K2) ).

% left_add_mult_distrib_mset
tff(fact_6771_union__fold__mset__add__mset,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5) = fold_mset(A,multiset(A),add_mset(A),A6,B5) ).

% union_fold_mset_add_mset
tff(fact_6772_comp__fun__commute_Ofold__mset__union,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),S2: B,M6: multiset(A),N7: multiset(A)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( fold_mset(A,B,F3,S2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M6),N7)) = fold_mset(A,B,F3,fold_mset(A,B,F3,S2,M6),N7) ) ) ).

% comp_fun_commute.fold_mset_union
tff(fact_6773_mset__subseteq__add__iff1,axiom,
    ! [A: $tType,J2: nat,I2: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J2),U)),N)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M)),N)) ) ) ).

% mset_subseteq_add_iff1
tff(fact_6774_mset__subseteq__add__iff2,axiom,
    ! [A: $tType,I2: nat,J2: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J2),U)),N)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N))) ) ) ).

% mset_subseteq_add_iff2
tff(fact_6775_sum__mset_Oeq__fold,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M6: multiset(A)] : comm_m7189776963980413722m_mset(A,M6) = fold_mset(A,A,plus_plus(A),zero_zero(A),M6) ) ).

% sum_mset.eq_fold
tff(fact_6776_sorted__list__of__multiset__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: multiset(A)] : linord6283353356039996273ltiset(A,M6) = fold_mset(A,list(A),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),nil(A),M6) ) ).

% sorted_list_of_multiset_def
tff(fact_6777_mult1__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : mult1(A,R3) = aa(fun(product_prod(multiset(A),multiset(A)),bool),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),bool)),fun(product_prod(multiset(A),multiset(A)),bool),product_case_prod(multiset(A),multiset(A),bool),aTP_Lamp_alc(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),bool)),R3))) ).

% mult1_def
tff(fact_6778_subset__mset_OcINF__union,axiom,
    ! [A: $tType,B: $tType,A6: set(B),F3: fun(B,multiset(A)),B5: set(B)] :
      ( ( A6 != bot_bot(set(B)) )
     => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
       => ( ( B5 != bot_bot(set(B)) )
         => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B5)))
           => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B5))) ) ) ) ) ) ).

% subset_mset.cINF_union
tff(fact_6779_subset__mset_Obdd__below__image__inf,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),G3: fun(B,multiset(A)),A6: set(B)] :
      ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_ald(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A6)))
    <=> ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
        & pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A6))) ) ) ).

% subset_mset.bdd_below_image_inf
tff(fact_6780_inter__mset__empty__distrib__left,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5)),C4) = zero_zero(multiset(A)) )
    <=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),C4) = zero_zero(multiset(A)) )
        & ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),B5),C4) = zero_zero(multiset(A)) ) ) ) ).

% inter_mset_empty_distrib_left
tff(fact_6781_inter__mset__empty__distrib__right,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4)) = zero_zero(multiset(A)) )
    <=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5) = zero_zero(multiset(A)) )
        & ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),C4) = zero_zero(multiset(A)) ) ) ) ).

% inter_mset_empty_distrib_right
tff(fact_6782_inter__subset__eq__union,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),B5))) ).

% inter_subset_eq_union
tff(fact_6783_inter__union__distrib__right,axiom,
    ! [A: $tType,C4: multiset(A),A6: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),A6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),B5)) ).

% inter_union_distrib_right
tff(fact_6784_inter__union__distrib__left,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),C4: 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)),inter_mset(A),A6),B5)),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4)) ).

% inter_union_distrib_left
tff(fact_6785_rel__fun__def,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A6: fun(A,fun(C,bool)),B5: fun(B,fun(D,bool)),X5: fun(A,B),Xa: fun(C,D)] :
      ( pp(aa(fun(C,D),bool,aa(fun(A,B),fun(fun(C,D),bool),bNF_rel_fun(A,C,B,D,A6,B5),X5),Xa))
    <=> ! [Xb5: A,Y4: C] :
          ( pp(aa(C,bool,aa(A,fun(C,bool),A6,Xb5),Y4))
         => pp(aa(D,bool,aa(B,fun(D,bool),B5,aa(A,B,X5,Xb5)),aa(C,D,Xa,Y4))) ) ) ).

% rel_fun_def
tff(fact_6786_rel__fun__eq__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: fun(B,fun(C,bool)),X5: fun(A,B),Xa: fun(A,C)] :
      ( pp(aa(fun(A,C),bool,aa(fun(A,B),fun(fun(A,C),bool),bNF_rel_fun(A,A,B,C,fequal(A),R2),X5),Xa))
    <=> ! [Xb5: A] : pp(aa(C,bool,aa(B,fun(C,bool),R2,aa(A,B,X5,Xb5)),aa(A,C,Xa,Xb5))) ) ).

% rel_fun_eq_rel
tff(fact_6787_multiset__union__diff__commute,axiom,
    ! [A: $tType,B5: multiset(A),C4: multiset(A),A6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),B5),C4) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),A6),B5)),C4) = 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)),minus_minus(multiset(A)),A6),C4)),B5) ) ) ).

% multiset_union_diff_commute
tff(fact_6788_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B),D3: B] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_ale(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),A6),B5),D3)))) ) ) ).

% finite_set_of_finite_funs
tff(fact_6789_Collect__all__eq,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_alf(fun(A,fun(B,bool)),fun(A,bool),P2)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ads(fun(A,fun(B,bool)),fun(B,set(A)),P2)),top_top(set(B)))) ).

% Collect_all_eq
tff(fact_6790_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P2: fun(A,bool)] : ord_Least(A,P2) = the(A,aTP_Lamp_alg(fun(A,bool),fun(A,bool),P2)) ) ).

% Least_def
tff(fact_6791_subset__mset_Omono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F3: fun(multiset(A),B),A6: multiset(A),B5: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(multiset(A),B,F3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(multiset(A),B,F3,A6)),aa(multiset(A),B,F3,B5)))) ) ) ).

% subset_mset.mono_inf
tff(fact_6792_subset__mset_OcINF__inf__distrib,axiom,
    ! [A: $tType,B: $tType,A6: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( ( A6 != bot_bot(set(B)) )
     => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
       => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A6)))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6))),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A6))) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_ald(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A6)) ) ) ) ) ).

% subset_mset.cINF_inf_distrib
tff(fact_6793_subset__mset_OcInf__union__distrib,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),A6))
       => ( ( B5 != bot_bot(set(multiset(A))) )
         => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),B5))
           => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),A6)),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),B5)) ) ) ) ) ) ).

% subset_mset.cInf_union_distrib
tff(fact_6794_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool)] : order_Greatest(A,P2) = the(A,aTP_Lamp_alh(fun(A,bool),fun(A,bool),P2)) ) ).

% Greatest_def
tff(fact_6795_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),P2: fun(A,bool)] : aa(fun(A,bool),A,least(A,Less_eq),P2) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ali(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P2)) ).

% ord.Least_def
tff(fact_6796_GreatestI__nat,axiom,
    ! [P2: fun(nat,bool),K2: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,P2,order_Greatest(nat,P2))) ) ) ).

% GreatestI_nat
tff(fact_6797_Greatest__le__nat,axiom,
    ! [P2: fun(nat,bool),K2: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),order_Greatest(nat,P2))) ) ) ).

% Greatest_le_nat
tff(fact_6798_GreatestI__ex__nat,axiom,
    ! [P2: fun(nat,bool),B2: nat] :
      ( ? [X_13: nat] : pp(aa(nat,bool,P2,X_13))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,P2,order_Greatest(nat,P2))) ) ) ).

% GreatestI_ex_nat
tff(fact_6799_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_6800_subset__mset_OLeast__def,axiom,
    ! [A: $tType,P2: fun(multiset(A),bool)] : aa(fun(multiset(A),bool),multiset(A),least(multiset(A),subseteq_mset(A)),P2) = the(multiset(A),aTP_Lamp_alj(fun(multiset(A),bool),fun(multiset(A),bool),P2)) ).

% subset_mset.Least_def
tff(fact_6801_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( order_Greatest(A,P2) = X ) ) ) ) ).

% Greatest_equality
tff(fact_6802_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A,Q2: fun(A,bool)] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P2,X3))
                 => ( ! [Y5: A] :
                        ( pp(aa(A,bool,P2,Y5))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),X3)) )
                   => pp(aa(A,bool,Q2,X3)) ) )
             => pp(aa(A,bool,Q2,order_Greatest(A,P2))) ) ) ) ) ).

% GreatestI2_order
tff(fact_6803_transfer__bforall__def,axiom,
    ! [A: $tType,X5: fun(A,bool),Xa: fun(A,bool)] :
      ( transfer_bforall(A,X5,Xa)
    <=> ! [Xb5: A] :
          ( pp(aa(A,bool,X5,Xb5))
         => pp(aa(A,bool,Xa,Xb5)) ) ) ).

% transfer_bforall_def
tff(fact_6804_max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : max_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_alk(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R2))) ).

% max_ext_eq
tff(fact_6805_bex__empty,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ~ ? [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),bot_bot(set(A))))
          & pp(aa(A,bool,P2,X5)) ) ).

% bex_empty
tff(fact_6806_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A6: set(A),Q2: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_all(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A6),Q2))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q2),X4)))) ) ) ) ).

% finite_Collect_bex
tff(fact_6807_bex__UNIV,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),top_top(set(A))))
          & pp(aa(A,bool,P2,X4)) )
    <=> ? [X_12: A] : pp(aa(A,bool,P2,X_12)) ) ).

% bex_UNIV
tff(fact_6808_Image__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P2: fun(B,fun(A,bool)),A6: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P2))),A6) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_alm(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),P2),A6)) ).

% Image_Collect_case_prod
tff(fact_6809_SUP__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_aln(set(A),fun(fun(A,bool),bool)) = bex(A) ).

% SUP_bool_eq
tff(fact_6810_Bex__fold,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
            & pp(aa(A,bool,P2,X4)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_alo(fun(A,bool),fun(A,fun(bool,bool)),P2),fFalse,A6)) ) ) ).

% Bex_fold
tff(fact_6811_Image__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(A)] : aa(set(A),set(B),image(A,B,R3),S2) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_alp(set(product_prod(A,B)),fun(set(A),fun(B,bool)),R3),S2)) ).

% Image_def
tff(fact_6812_Union__eq,axiom,
    ! [A: $tType,A6: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_alq(set(set(A)),fun(A,bool),A6)) ).

% Union_eq
tff(fact_6813_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),aa(set(A),set(B),image2(A,B,F3),A6)) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_alr(fun(A,B),fun(set(A),fun(A,bool)),F3),A6)) ).

% vimage_image_eq
tff(fact_6814_image__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A)] : aa(set(A),set(B),image2(A,B,F3),A6) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_als(fun(A,B),fun(set(A),fun(B,bool)),F3),A6)) ).

% image_def
tff(fact_6815_UNION__eq,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6)) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_alt(fun(B,set(A)),fun(set(B),fun(A,bool)),B5),A6)) ).

% UNION_eq
tff(fact_6816_Collect__bex__eq,axiom,
    ! [A: $tType,B: $tType,A6: set(B),P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_alu(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),A6),P2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ads(fun(A,fun(B,bool)),fun(B,set(A)),P2)),A6)) ).

% Collect_bex_eq
tff(fact_6817_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A6: set(nat),B5: set(nat)] : nths(A,nths(A,Xs,A6),B5) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_alw(set(nat),fun(set(nat),fun(nat,bool)),A6),B5))) ).

% nths_nths
tff(fact_6818_max__extp_Ocases,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),A1: set(A),A22: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R2),A1),A22))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
         => ( pp(aa(set(A),bool,finite_finite2(A),A22))
           => ( ( A22 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
             => ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),A1))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),A22))
                        & pp(aa(A,bool,aa(A,fun(A,bool),R2,X5),Xa4)) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_6819_max__extp_Osimps,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),A1: set(A),A22: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R2),A1),A22))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
        & pp(aa(set(A),bool,finite_finite2(A),A22))
        & ( A22 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A1))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),A22))
                & pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),Xa3)) ) ) ) ) ).

% max_extp.simps
tff(fact_6820_max__extp_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),R2: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
       => ( ( Y6 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,member(A,Xa),Y6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Xa)) ) )
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R2),X6),Y6)) ) ) ) ) ).

% max_extp.max_extI
tff(fact_6821_map__project__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),A6: set(A)] : map_project(A,B,F3,A6) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_alx(fun(A,option(B)),fun(set(A),fun(B,bool)),F3),A6)) ).

% map_project_def
tff(fact_6822_subset__mset_OGreatest__def,axiom,
    ! [A: $tType,P2: fun(multiset(A),bool)] : aa(fun(multiset(A),bool),multiset(A),greatest(multiset(A),subseteq_mset(A)),P2) = the(multiset(A),aTP_Lamp_aly(fun(multiset(A),bool),fun(multiset(A),bool),P2)) ).

% subset_mset.Greatest_def
tff(fact_6823_order_OGreatest_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : greatest(A,Less_eq) = greatest(A,Less_eq) ).

% order.Greatest.cong
tff(fact_6824_multp__code__def,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( multp_code(A,P2,N7,M6)
    <=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)) != zero_zero(multiset(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),N7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
                & pp(aa(A,bool,aa(A,fun(A,bool),P2,X4),Xa3)) ) ) ) ) ).

% multp_code_def
tff(fact_6825_min__ext__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : min_ext(A,R3) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_alz(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R3)) ).

% min_ext_def
tff(fact_6826_ball__empty,axiom,
    ! [A: $tType,P2: fun(A,bool),X5: A] :
      ( pp(aa(set(A),bool,member(A,X5),bot_bot(set(A))))
     => pp(aa(A,bool,P2,X5)) ) ).

% ball_empty
tff(fact_6827_ball__UNIV,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),top_top(set(A))))
         => pp(aa(A,bool,P2,X4)) )
    <=> ! [X_12: A] : pp(aa(A,bool,P2,X_12)) ) ).

% ball_UNIV
tff(fact_6828_INF__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_ama(set(A),fun(fun(A,bool),bool)) = ball(A) ).

% INF_bool_eq
tff(fact_6829_Collect__ball__eq,axiom,
    ! [A: $tType,B: $tType,A6: set(B),P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_amb(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),A6),P2)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ads(fun(A,fun(B,bool)),fun(B,set(A)),P2)),A6)) ).

% Collect_ball_eq
tff(fact_6830_INTER__eq,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A6: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A6)) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_amc(fun(B,set(A)),fun(set(B),fun(A,bool)),B5),A6)) ).

% INTER_eq
tff(fact_6831_Inter__eq,axiom,
    ! [A: $tType,A6: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_amd(set(set(A)),fun(A,bool),A6)) ).

% Inter_eq
tff(fact_6832_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A6: fun(B,set(C)),F3: fun(C,bool),G3: fun(A,B),X5: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),comp(B,bool,A,aa(fun(C,bool),fun(B,bool),aTP_Lamp_ame(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),A6),F3)),G3),X5))
    <=> ! [Xa3: C] :
          ( pp(aa(set(C),bool,member(C,Xa3),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),comp(B,set(C),A,A6),G3),X5)))
         => pp(aa(C,bool,F3,Xa3)) ) ) ).

% Ball_comp_iff
tff(fact_6833_Ball__Collect,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A6))
         => pp(aa(A,bool,P2,X4)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(fun(A,bool),set(A),collect(A),P2))) ) ).

% Ball_Collect
tff(fact_6834_irrefl__distinct,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( irrefl(A,R3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_amf(A,fun(A,bool))),X4)) ) ) ).

% irrefl_distinct
tff(fact_6835_Field__not__elem,axiom,
    ! [A: $tType,V2: A,R2: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,member(A,V2),field2(A,R2)))
     => ! [X5: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),R2))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_amg(A,fun(A,fun(A,bool)),V2)),X5)) ) ) ).

% Field_not_elem
tff(fact_6836_congruent__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F3: fun(A,B)] :
      ( equiv_congruent(A,B,R3,F3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ahp(fun(A,B),fun(A,fun(A,bool)),F3)),X4)) ) ) ).

% congruent_def
tff(fact_6837_congruent2__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( equiv_congruent2(A,B,C,R12,R23,F3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R12))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,fun(B,C)),fun(A,fun(A,bool)),aTP_Lamp_ami(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),R23),F3)),X4)) ) ) ).

% congruent2_def
tff(fact_6838_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A6) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_amj(set(A),fun(A,bool),A6))) ) ).

% Inf_eq_Sup
tff(fact_6839_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A6) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_amk(set(A),fun(A,bool),A6))) ) ).

% Sup_eq_Inf
tff(fact_6840_trans__join,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( trans(A,R3)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_amm(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)),X4)) ) ) ).

% trans_join
tff(fact_6841_Ball__fold,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => pp(aa(A,bool,P2,X4)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_amn(fun(A,bool),fun(A,fun(bool,bool)),P2),fTrue,A6)) ) ) ).

% Ball_fold
tff(fact_6842_takeWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool),Ys2: list(A)] :
      ( ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P2,X3)) )
       => ( takeWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P2,Ys2)) ) )
      & ( ~ ! [X5: A] :
              ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P2,X5)) )
       => ( takeWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = takeWhile(A,P2,Xs) ) ) ) ).

% takeWhile_append
tff(fact_6843_dropWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool),Ys2: list(A)] :
      ( ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P2,X3)) )
       => ( dropWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = dropWhile(A,P2,Ys2) ) )
      & ( ~ ! [X5: A] :
              ( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P2,X5)) )
       => ( dropWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P2,Xs)),Ys2) ) ) ) ).

% dropWhile_append
tff(fact_6844_Union__maximal__sets,axiom,
    ! [A: $tType,F13: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F13))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_amo(set(set(A)),fun(set(A),bool),F13))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F13) ) ) ).

% Union_maximal_sets
tff(fact_6845_Sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A6: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_amp(set(set(A)),fun(set(A),bool),A6)))) ) ).

% Sup_Inf
tff(fact_6846_Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A6: set(set(A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_amp(set(set(A)),fun(set(A),bool),A6)))) ) ).

% Inf_Sup
tff(fact_6847_Func__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] : bNF_Wellorder_Func(A,B,A6,B5) = aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aa(set(B),fun(fun(A,B),bool),aTP_Lamp_amq(set(A),fun(set(B),fun(fun(A,B),bool)),A6),B5)) ).

% Func_def
tff(fact_6848_Inf__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(fun(filter(A),bool),set(filter(A)),collect(filter(A)),aTP_Lamp_amr(set(filter(A)),fun(filter(A),bool),S))) ).

% Inf_filter_def
tff(fact_6849_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys2))))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),fequal(A)),X4)) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_6850_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ams(set(set(A)),fun(set(A),bool),A6)))))) ) ).

% finite_Inf_Sup
tff(fact_6851_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_amt(set(set(A)),fun(set(A),bool),A6))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A6)))) ) ).

% Sup_Inf_le
tff(fact_6852_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_amp(set(set(A)),fun(set(A),bool),A6)))))) ) ).

% Inf_Sup_le
tff(fact_6853_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G3: fun(B,A),A6: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_amu(fun(B,A),fun(set(B),A),G3)),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_amv(fun(B,A),fun(set(B),A),G3)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_amw(set(set(B)),fun(set(B),bool),A6)))) ) ).

% INF_SUP_set
tff(fact_6854_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G3: fun(B,A),A6: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_amv(fun(B,A),fun(set(B),A),G3)),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_amu(fun(B,A),fun(set(B),A),G3)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_amw(set(set(B)),fun(set(B),bool),A6)))) ) ).

% SUP_INF_set
tff(fact_6855_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys2) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ! [X3: product_prod(list(A),list(A))] :
              ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X3),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys2))))
             => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_amx(list(A),fun(list(A),bool))),X3)) )
         => ( Xs = Ys2 ) ) ) ) ).

% concat_injective
tff(fact_6856_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys2: list(list(A))] :
      ( ! [X3: product_prod(list(A),list(A))] :
          ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X3),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys2))))
         => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_amx(list(A),fun(list(A),bool))),X3)) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys2) )
       => ( ( concat(A,Xs) = concat(A,Ys2) )
        <=> ( Xs = Ys2 ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_6857_option__Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A6: set(set(option(A)))] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),complete_Sup_Sup(option(A))),A6))),aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),complete_Inf_Inf(option(A))),aa(fun(set(option(A)),bool),set(set(option(A))),collect(set(option(A))),aTP_Lamp_amy(set(set(option(A))),fun(set(option(A)),bool),A6)))))) ) ).

% option_Inf_Sup
tff(fact_6858_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2),X4)) ) ) ) ).

% list_all2_iff
tff(fact_6859_Sup__int__def,axiom,
    ! [X6: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X6) = the(int,aTP_Lamp_amz(set(int),fun(int,bool),X6)) ).

% Sup_int_def
tff(fact_6860_subset__mset_OcSup__cInf,axiom,
    ! [A: $tType,S: set(multiset(A))] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),S))
       => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aTP_Lamp_ana(set(multiset(A)),fun(multiset(A),bool),S))) ) ) ) ).

% subset_mset.cSup_cInf
tff(fact_6861_subset__mset_OcInf__cSup,axiom,
    ! [A: $tType,S: set(multiset(A))] :
      ( ( S != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),S))
       => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),S) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aTP_Lamp_anb(set(multiset(A)),fun(multiset(A),bool),S))) ) ) ) ).

% subset_mset.cInf_cSup
tff(fact_6862_multeqp__code__def,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),N7: multiset(A),M6: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multeqp_code(A,P2),N7),M6))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),N7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
         => ? [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),M6),N7)))))
              & pp(aa(A,bool,aa(A,fun(A,bool),P2,X4),Xa3)) ) ) ) ).

% multeqp_code_def
tff(fact_6863_Under__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] : order_Under(A,R3,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_anc(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A6)) ).

% Under_def
tff(fact_6864_UnderS__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] : order_UnderS(A,R3,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_and(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A6)) ).

% UnderS_def
tff(fact_6865_Above__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] : order_Above(A,R3,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ane(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A6)) ).

% Above_def
tff(fact_6866_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Xs),Ys2)),listrel(A,B,R3)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),X4)) ) ) ) ).

% listrel_iff_zip
tff(fact_6867_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S2))
     => pp(aa(set(product_prod(list(A),list(B))),bool,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),bool),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R3)),listrel(A,B,S2))) ) ).

% listrel_mono
tff(fact_6868_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys2: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2))),listrel(A,B,R3)))
     => ~ ! [X3: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
           => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y)),R3))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Xs2),Ys2)),listrel(A,B,R3))) ) ) ) ).

% listrel_Cons2
tff(fact_6869_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys2: list(A),Xs: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)),Xs)),listrel(A,B,R3)))
     => ~ ! [Y3: B,Ys5: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5) )
           => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y3)),R3))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Ys2),Ys5)),listrel(A,B,R3))) ) ) ) ).

% listrel_Cons1
tff(fact_6870_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,R3: set(product_prod(A,B)),Xs: list(A),Ys2: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,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)),R3))
     => ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Xs),Ys2)),listrel(A,B,R3)))
       => pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2))),listrel(A,B,R3))) ) ) ).

% listrel.Cons
tff(fact_6871_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),A1),A22)),listrel(A,B,R3)))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X3: A,Y3: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Ys5: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5) )
                 => ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R3))
                   => ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Xs2),Ys5)),listrel(A,B,R3))) ) ) ) ) ) ).

% listrel.cases
tff(fact_6872_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),A1),A22)),listrel(A,B,R3)))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X4: A,Y4: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
            & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4)),R3))
            & pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Xs3),Ys4)),listrel(A,B,R3))) ) ) ) ).

% listrel.simps
tff(fact_6873_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),transitive_rtrancl(list(A),listrel1(A,R3)))) ).

% listrel_subset_rtrancl_listrel1
tff(fact_6874_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),Xs),Ys2)),listrel(A,B,R3)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N3)),aa(nat,B,nth(B,Ys2),N3))),R3)) ) ) ) ).

% listrel_iff_nth
tff(fact_6875_listrel__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : listrel(A,B,X5) = aa(fun(product_prod(list(A),list(B)),bool),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),bool)),fun(product_prod(list(A),list(B)),bool),product_case_prod(list(A),list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)))) ).

% listrel_def
tff(fact_6876_listrel1__subset__listrel,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R6))
     => ( refl_on(A,top_top(set(A)),R6)
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel(A,A,R6))) ) ) ).

% listrel1_subset_listrel
tff(fact_6877_refl__on__Un,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),B5: set(A),S2: set(product_prod(A,A))] :
      ( refl_on(A,A6,R3)
     => ( refl_on(A,B5,S2)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5),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))),R3),S2)) ) ) ).

% refl_on_Un
tff(fact_6878_refl__on__domain,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,A6,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( pp(aa(set(A),bool,member(A,A3),A6))
          & pp(aa(set(A),bool,member(A,B2),A6)) ) ) ) ).

% refl_on_domain
tff(fact_6879_refl__onD2,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A6,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => pp(aa(set(A),bool,member(A,Y),A6)) ) ) ).

% refl_onD2
tff(fact_6880_refl__onD1,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A6,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
       => pp(aa(set(A),bool,member(A,X),A6)) ) ) ).

% refl_onD1
tff(fact_6881_refl__onD,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,A6,R3)
     => ( pp(aa(set(A),bool,member(A,A3),A6))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R3)) ) ) ).

% refl_onD
tff(fact_6882_refl__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : refl_on(A,top_top(set(A)),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))),R3),id2(A))) ).

% refl_reflcl
tff(fact_6883_refl__on__comp__subset,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A6,R3)
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),relcomp(A,A,A,converse(A,A,R3),R3))) ) ).

% refl_on_comp_subset
tff(fact_6884_refl__onI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R3)) )
       => refl_on(A,A6,R3) ) ) ).

% refl_onI
tff(fact_6885_refl__on__def,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A6,R3)
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R3)) ) ) ) ).

% refl_on_def
tff(fact_6886_Refl__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( refl_on(A,field2(A,R3),R3)
     => refl_on(A,field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% Refl_Restr
tff(fact_6887_refl__on__def_H,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A6,R3)
    <=> ( ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),R3))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_anf(set(A),fun(A,fun(A,bool)),A6)),X4)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R3)) ) ) ) ).

% refl_on_def'
tff(fact_6888_Image__absorb__rtrancl,axiom,
    ! [A: $tType,A6: set(product_prod(A,A)),B5: set(A),C4: set(A)] :
      ( trans(A,A6)
     => ( refl_on(A,B5,A6)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B5))
         => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,A6)),C4) = aa(set(A),set(A),image(A,A,A6),C4) ) ) ) ) ).

% Image_absorb_rtrancl
tff(fact_6889_refl__on__reflcl__Image,axiom,
    ! [A: $tType,B5: set(A),A6: set(product_prod(A,A)),C4: set(A)] :
      ( refl_on(A,B5,A6)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),B5))
       => ( aa(set(A),set(A),image(A,A,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))),A6),id2(A))),C4) = aa(set(A),set(A),image(A,A,A6),C4) ) ) ) ).

% refl_on_reflcl_Image
tff(fact_6890_Refl__Field__Restr2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( refl_on(A,field2(A,R3),R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),field2(A,R3)))
       => ( field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) = A6 ) ) ) ).

% Refl_Field_Restr2
tff(fact_6891_Refl__Field__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( refl_on(A,field2(A,R3),R3)
     => ( field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),field2(A,R3)),A6) ) ) ).

% Refl_Field_Restr
tff(fact_6892_refl__on__singleton,axiom,
    ! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),X),X)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_6893_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),X5: list(A),Xa: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),X5),Xa))
    <=> pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),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)),X5),Xa)),listrel(A,B,R3))) ) ).

% listrelp_listrel_eq
tff(fact_6894_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] : pp(aa(fun(fun(A,bool),fun(A,bool)),bool,order_mono(fun(A,bool),fun(A,bool)),aTP_Lamp_ang(fun(A,A),fun(fun(A,bool),fun(A,bool)),F3))) ) ).

% iteratesp.mono
tff(fact_6895_repeat__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa2: nat] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),X),X))
     => ( aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),Xa2),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(nat,fun(A,nat),aTP_Lamp_anh(fun(A,nat),fun(nat,fun(A,nat)),X),Xa2)) ) ) ).

% repeat_mset.abs_eq
tff(fact_6896_eq__onp__True,axiom,
    ! [A: $tType] : bNF_eq_onp(A,aTP_Lamp_av(A,bool)) = fequal(A) ).

% eq_onp_True
tff(fact_6897_eq__onp__def,axiom,
    ! [A: $tType,R2: fun(A,bool),X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,R2),X5),Xa))
    <=> ( pp(aa(A,bool,R2,X5))
        & ( X5 = Xa ) ) ) ).

% eq_onp_def
tff(fact_6898_chain__compr,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),A6: set(A),P2: fun(A,bool)] :
      ( comple1602240252501008431_chain(A,Ord,A6)
     => comple1602240252501008431_chain(A,Ord,aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2))) ) ).

% chain_compr
tff(fact_6899_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A6: set(A),X: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A6)
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ).

% ccpo_Sup_upper
tff(fact_6900_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A6: set(A),Z4: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A6)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),Z4)) ) ) ) ).

% ccpo_Sup_least
tff(fact_6901_ccpo__Sup__mono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A6: set(A),B5: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A6)
         => ( comple1602240252501008431_chain(A,ord_less_eq(A),B5)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A6))
                 => ? [Xa: A] :
                      ( pp(aa(set(A),bool,member(A,Xa),B5))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ) ) ).

% ccpo_Sup_mono
tff(fact_6902_chain__subset,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),A6: set(A),B5: set(A)] :
      ( comple1602240252501008431_chain(A,Ord,A6)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
       => comple1602240252501008431_chain(A,Ord,B5) ) ) ).

% chain_subset
tff(fact_6903_eq__onp__le__eq,axiom,
    ! [A: $tType,P2: fun(A,bool)] : pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),bNF_eq_onp(A,P2)),fequal(A))) ).

% eq_onp_le_eq
tff(fact_6904_eq__onp__mono__iff,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),bNF_eq_onp(A,P2)),bNF_eq_onp(A,Q2)))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Q2)) ) ).

% eq_onp_mono_iff
tff(fact_6905_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_6906_rel__fun__eq__eq__onp,axiom,
    ! [A: $tType,B: $tType,P2: fun(B,bool)] : bNF_rel_fun(A,A,B,B,fequal(A),bNF_eq_onp(B,P2)) = bNF_eq_onp(fun(A,B),aTP_Lamp_ani(fun(B,bool),fun(fun(A,B),bool),P2)) ).

% rel_fun_eq_eq_onp
tff(fact_6907_rel__fun__eq__onp__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R2: fun(A,bool),S: fun(B,fun(C,bool)),X5: fun(A,B),Xa: fun(A,C)] :
      ( pp(aa(fun(A,C),bool,aa(fun(A,B),fun(fun(A,C),bool),bNF_rel_fun(A,A,B,C,bNF_eq_onp(A,R2),S),X5),Xa))
    <=> ! [Xb5: A] :
          ( pp(aa(A,bool,R2,Xb5))
         => pp(aa(C,bool,aa(B,fun(C,bool),S,aa(A,B,X5,Xb5)),aa(A,C,Xa,Xb5))) ) ) ).

% rel_fun_eq_onp_rel
tff(fact_6908_filter__mset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),bool,aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),bool),bNF_rel_fun(fun(A,bool),fun(A,bool),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(fun(A,bool)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)))),aTP_Lamp_anj(fun(A,bool),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_anj(fun(A,bool),fun(fun(A,nat),fun(A,nat))))) ).

% filter_mset.rsp
tff(fact_6909_zero__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),aTP_Lamp_akw(A,nat)),aTP_Lamp_akw(A,nat))) ).

% zero_multiset.rsp
tff(fact_6910_add__mset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(A,fun(fun(A,nat),fun(A,nat))),bool,aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(fun(A,fun(fun(A,nat),fun(A,nat))),bool),bNF_rel_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(A),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)))),aTP_Lamp_akz(A,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_akz(A,fun(fun(A,nat),fun(A,nat))))) ).

% add_mset.rsp
tff(fact_6911_plus__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)))),aTP_Lamp_akx(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_akx(fun(A,nat),fun(fun(A,nat),fun(A,nat))))) ).

% plus_multiset.rsp
tff(fact_6912_minus__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)))),aTP_Lamp_aky(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_aky(fun(A,nat),fun(fun(A,nat),fun(A,nat))))) ).

% minus_multiset.rsp
tff(fact_6913_repeat__mset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(nat,fun(fun(A,nat),fun(A,nat))),bool,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(fun(A,nat),fun(A,nat))),bool),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),fequal(nat),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)))),aTP_Lamp_ala(nat,fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_ala(nat,fun(fun(A,nat),fun(A,nat))))) ).

% repeat_mset.rsp
tff(fact_6914_add__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa2: A] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),X),X))
     => ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Xa2),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(A,fun(A,nat),aTP_Lamp_ank(fun(A,nat),fun(A,fun(A,nat)),X),Xa2)) ) ) ).

% add_mset.abs_eq
tff(fact_6915_plus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa2: fun(A,nat),X: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),Xa2),Xa2))
     => ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),X),X))
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(fun(A,nat),multiset(A),abs_multiset(A),Xa2)),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_akx(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa2),X)) ) ) ) ).

% plus_multiset.abs_eq
tff(fact_6916_minus__multiset_Oabs__eq,axiom,
    ! [A: $tType,Xa2: fun(A,nat),X: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),Xa2),Xa2))
     => ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),X),X))
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(fun(A,nat),multiset(A),abs_multiset(A),Xa2)),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_aky(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa2),X)) ) ) ) ).

% minus_multiset.abs_eq
tff(fact_6917_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A6: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A6)
         => ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => pp(aa(set(A),bool,member(A,aa(set(A),A,complete_Sup_Sup(A),A6)),A6)) ) ) ) ) ).

% in_chain_finite
tff(fact_6918_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X5: fun(A,A)] : comple7512665784863727008ratesp(A,X5) = complete_lattice_lfp(fun(A,bool),aTP_Lamp_ang(fun(A,A),fun(fun(A,bool),fun(A,bool)),X5)) ) ).

% iteratesp_def
tff(fact_6919_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M6: set(A),F3: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M6)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),M6))
               => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X3)) )
           => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),aa(set(A),A,complete_Sup_Sup(A),M6))) ) ) ) ).

% iteratesp.Sup
tff(fact_6920_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A3: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),A3))
         => ( ! [X3: A] :
                ( ( A3 = aa(A,A,F3,X3) )
               => ~ pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X3)) )
           => ~ ! [M11: set(A)] :
                  ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M11) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M11)
                   => ~ ! [X5: A] :
                          ( pp(aa(set(A),bool,member(A,X5),M11))
                         => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X5)) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_6921_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A3: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),A3))
        <=> ( ? [X4: A] :
                ( ( A3 = aa(A,A,F3,X4) )
                & pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X4)) )
            | ? [M13: set(A)] :
                ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M13) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M13)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),M13))
                   => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X4)) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_6922_Inf__multiset_Oabs__eq,axiom,
    ! [A: $tType,X: set(fun(A,nat))] :
      ( pp(aa(set(fun(A,nat)),bool,aa(set(fun(A,nat)),fun(set(fun(A,nat)),bool),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool))),X),X))
     => ( aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(fun(A,nat)),set(multiset(A)),image2(fun(A,nat),multiset(A),abs_multiset(A)),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_aku(set(fun(A,nat)),fun(A,nat)),X)) ) ) ).

% Inf_multiset.abs_eq
tff(fact_6923_flat__lub__def,axiom,
    ! [A: $tType,A6: set(A),B2: A] :
      ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A6) = B2 ) )
      & ( ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A6) = the(A,aa(A,fun(A,bool),aTP_Lamp_anl(set(A),fun(A,fun(A,bool)),A6),B2)) ) ) ) ).

% flat_lub_def
tff(fact_6924_rel__set__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X5: set(A),Xa: set(B)] :
      ( pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,R2),X5),Xa))
    <=> ( ! [Xb5: A] :
            ( pp(aa(set(A),bool,member(A,Xb5),X5))
           => ? [Xc2: B] :
                ( pp(aa(set(B),bool,member(B,Xc2),Xa))
                & pp(aa(B,bool,aa(A,fun(B,bool),R2,Xb5),Xc2)) ) )
        & ! [Xb5: B] :
            ( pp(aa(set(B),bool,member(B,Xb5),Xa))
           => ? [Xc2: A] :
                ( pp(aa(set(A),bool,member(A,Xc2),X5))
                & pp(aa(B,bool,aa(A,fun(B,bool),R2,Xc2),Xb5)) ) ) ) ) ).

% rel_set_def
tff(fact_6925_fun_Oset__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(fun(D,B),set(B)),bool,aa(fun(fun(D,A),set(A)),fun(fun(fun(D,B),set(B)),bool),bNF_rel_fun(fun(D,A),fun(D,B),set(A),set(B),bNF_rel_fun(D,D,A,B,fequal(D),R2),bNF_rel_set(A,B,R2)),aTP_Lamp_anm(fun(D,A),set(A))),aTP_Lamp_ann(fun(D,B),set(B)))) ).

% fun.set_transfer
tff(fact_6926_Inf__multiset_Orsp,axiom,
    ! [A: $tType] : pp(aa(fun(set(fun(A,nat)),fun(A,nat)),bool,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(fun(A,nat)),fun(A,nat)),bool),bNF_rel_fun(set(fun(A,nat)),set(fun(A,nat)),fun(A,nat),fun(A,nat),bNF_rel_set(fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool))),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool))),aTP_Lamp_aku(set(fun(A,nat)),fun(A,nat))),aTP_Lamp_aku(set(fun(A,nat)),fun(A,nat)))) ).

% Inf_multiset.rsp
tff(fact_6927_INF__parametric,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [A6: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(fun(B,C),C)),bool,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,A6),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,A6,fequal(C)),fequal(C))),aTP_Lamp_ano(set(A),fun(fun(A,C),C))),aTP_Lamp_anp(set(B),fun(fun(B,C),C)))) ) ).

% INF_parametric
tff(fact_6928_SUP__parametric,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [R2: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(fun(B,C),C)),bool,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,R2),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,R2,fequal(C)),fequal(C))),aTP_Lamp_anq(set(A),fun(fun(A,C),C))),aTP_Lamp_anr(set(B),fun(fun(B,C),C)))) ) ).

% SUP_parametric
tff(fact_6929_union__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(set(B),set(B))),bool,aa(fun(set(A),fun(set(A),set(A))),fun(fun(set(B),fun(set(B),set(B))),bool),bNF_rel_fun(set(A),set(B),fun(set(A),set(A)),fun(set(B),set(B)),bNF_rel_set(A,B,A6),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A6),bNF_rel_set(A,B,A6))),sup_sup(set(A))),sup_sup(set(B)))) ).

% union_transfer
tff(fact_6930_rel__set__mono,axiom,
    ! [B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A6),B5))
     => pp(aa(fun(set(A),fun(set(B),bool)),bool,aa(fun(set(A),fun(set(B),bool)),fun(fun(set(A),fun(set(B),bool)),bool),ord_less_eq(fun(set(A),fun(set(B),bool))),bNF_rel_set(A,B,A6)),bNF_rel_set(A,B,B5))) ) ).

% rel_set_mono
tff(fact_6931_Union__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] : pp(aa(fun(set(set(B)),set(B)),bool,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),bool),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A6)),bNF_rel_set(A,B,A6)),complete_Sup_Sup(set(A))),complete_Sup_Sup(set(B)))) ).

% Union_transfer
tff(fact_6932_set__relator__eq__onp,axiom,
    ! [A: $tType,P2: fun(A,bool)] : bNF_rel_set(A,A,bNF_eq_onp(A,P2)) = bNF_eq_onp(set(A),aTP_Lamp_ans(fun(A,bool),fun(set(A),bool),P2)) ).

% set_relator_eq_onp
tff(fact_6933_UNION__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] : pp(aa(fun(set(B),fun(fun(B,set(D)),set(D))),bool,aa(fun(set(A),fun(fun(A,set(C)),set(C))),fun(fun(set(B),fun(fun(B,set(D)),set(D))),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,set(C)),set(C)),fun(fun(B,set(D)),set(D)),bNF_rel_set(A,B,A6),bNF_rel_fun(fun(A,set(C)),fun(B,set(D)),set(C),set(D),bNF_rel_fun(A,B,set(C),set(D),A6,bNF_rel_set(C,D,B5)),bNF_rel_set(C,D,B5))),aTP_Lamp_ant(set(A),fun(fun(A,set(C)),set(C)))),aTP_Lamp_anu(set(B),fun(fun(B,set(D)),set(D))))) ).

% UNION_transfer
tff(fact_6934_Inf__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(set(multiset(A)),multiset(A)),bool,aa(fun(set(fun(A,nat)),fun(A,nat)),fun(fun(set(multiset(A)),multiset(A)),bool),bNF_rel_fun(set(fun(A,nat)),set(multiset(A)),fun(A,nat),multiset(A),bNF_rel_set(fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A))),pcr_multiset(A,A,fequal(A))),aTP_Lamp_aku(set(fun(A,nat)),fun(A,nat))),complete_Inf_Inf(multiset(A)))) ).

% Inf_multiset.transfer
tff(fact_6935_filter__mset_Oabs__eq,axiom,
    ! [A: $tType,X: fun(A,nat),Xa2: fun(A,bool)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),X),X))
     => ( aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),Xa2),aa(fun(A,nat),multiset(A),abs_multiset(A),X)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,bool),fun(A,nat),aTP_Lamp_anv(fun(A,nat),fun(fun(A,bool),fun(A,nat)),X),Xa2)) ) ) ).

% filter_mset.abs_eq
tff(fact_6936_filter__mset__True,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_av(A,bool)),M6) = M6 ).

% filter_mset_True
tff(fact_6937_filter__union__mset,axiom,
    ! [A: $tType,P2: fun(A,bool),M6: multiset(A),N7: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),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)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),M6)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),N7)) ).

% filter_union_mset
tff(fact_6938_union__filter__mset__complement,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool),M6: multiset(A)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
        <=> ~ pp(aa(A,bool,Q2,X3)) )
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),M6)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),Q2),M6)) = M6 ) ) ).

% union_filter_mset_complement
tff(fact_6939_filter__mset__False,axiom,
    ! [A: $tType,M6: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_aq(A,bool)),M6) = zero_zero(multiset(A)) ).

% filter_mset_False
tff(fact_6940_set__mset__filter,axiom,
    ! [A: $tType,P2: fun(A,bool),M6: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),M6)) = aa(fun(A,bool),set(A),collect(A),aa(multiset(A),fun(A,bool),aTP_Lamp_anw(fun(A,bool),fun(multiset(A),fun(A,bool)),P2),M6)) ).

% set_mset_filter
tff(fact_6941_mset__filter,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : mset(A,filter2(A,P2,Xs)) = aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),mset(A,Xs)) ).

% mset_filter
tff(fact_6942_filter__mset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(fun(A,bool),fun(multiset(A),multiset(A))),bool,aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,bool),fun(multiset(A),multiset(A))),bool),bNF_rel_fun(fun(A,bool),fun(A,bool),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(fun(A,bool)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_anj(fun(A,bool),fun(fun(A,nat),fun(A,nat)))),filter_mset(A))) ).

% filter_mset.transfer
tff(fact_6943_image__mset__If,axiom,
    ! [A: $tType,B: $tType,P2: fun(B,bool),F3: fun(B,A),G3: fun(B,A),A6: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_vf(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P2),F3),G3)),A6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(B),multiset(A),image_mset(B,A,F3),aa(multiset(B),multiset(B),aa(fun(B,bool),fun(multiset(B),multiset(B)),filter_mset(B),P2),A6))),aa(multiset(B),multiset(A),image_mset(B,A,G3),aa(multiset(B),multiset(B),aa(fun(B,bool),fun(multiset(B),multiset(B)),filter_mset(B),aTP_Lamp_vg(fun(B,bool),fun(B,bool),P2)),A6))) ).

% image_mset_If
tff(fact_6944_filter__filter__mset,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool),M6: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),Q2),M6)) = aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_qi(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2)),M6) ).

% filter_filter_mset
tff(fact_6945_multiset__partition,axiom,
    ! [A: $tType,M6: multiset(A),P2: fun(A,bool)] : M6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),M6)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2)),M6)) ).

% multiset_partition
tff(fact_6946_size__filter__mset__lesseq,axiom,
    ! [A: $tType,F3: fun(A,bool),M6: multiset(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),F3),M6))),aa(multiset(A),nat,size_size(multiset(A)),M6))) ).

% size_filter_mset_lesseq
tff(fact_6947_zero__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(multiset(A),bool,aa(fun(A,nat),fun(multiset(A),bool),pcr_multiset(A,A,fequal(A)),aTP_Lamp_akw(A,nat)),zero_zero(multiset(A)))) ).

% zero_multiset.transfer
tff(fact_6948_multiset_Orep__transfer,axiom,
    ! [D: $tType,E: $tType,T8: fun(D,fun(E,bool))] : pp(aa(fun(multiset(E),fun(E,nat)),bool,aa(fun(fun(D,nat),fun(D,nat)),fun(fun(multiset(E),fun(E,nat)),bool),bNF_rel_fun(fun(D,nat),multiset(E),fun(D,nat),fun(E,nat),pcr_multiset(D,E,T8),bNF_rel_fun(D,E,nat,nat,T8,fequal(nat))),aTP_Lamp_anx(fun(D,nat),fun(D,nat))),count(E))) ).

% multiset.rep_transfer
tff(fact_6949_add__mset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(A,fun(multiset(A),multiset(A))),bool,aa(fun(A,fun(fun(A,nat),fun(A,nat))),fun(fun(A,fun(multiset(A),multiset(A))),bool),bNF_rel_fun(A,A,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(A),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_akz(A,fun(fun(A,nat),fun(A,nat)))),add_mset(A))) ).

% add_mset.transfer
tff(fact_6950_plus__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(multiset(A),fun(multiset(A),multiset(A))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(multiset(A),fun(multiset(A),multiset(A))),bool),bNF_rel_fun(fun(A,nat),multiset(A),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),pcr_multiset(A,A,fequal(A)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_akx(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),plus_plus(multiset(A)))) ).

% plus_multiset.transfer
tff(fact_6951_minus__multiset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(multiset(A),fun(multiset(A),multiset(A))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(multiset(A),fun(multiset(A),multiset(A))),bool),bNF_rel_fun(fun(A,nat),multiset(A),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),pcr_multiset(A,A,fequal(A)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_aky(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),minus_minus(multiset(A)))) ).

% minus_multiset.transfer
tff(fact_6952_repeat__mset_Otransfer,axiom,
    ! [A: $tType] : pp(aa(fun(nat,fun(multiset(A),multiset(A))),bool,aa(fun(nat,fun(fun(A,nat),fun(A,nat))),fun(fun(nat,fun(multiset(A),multiset(A))),bool),bNF_rel_fun(nat,nat,fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),fequal(nat),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_ala(nat,fun(fun(A,nat),fun(A,nat)))),repeat_mset(A))) ).

% repeat_mset.transfer
tff(fact_6953_filter__mset__def,axiom,
    ! [A: $tType] : filter_mset(A) = aa(fun(fun(A,bool),fun(fun(A,nat),fun(A,nat))),fun(fun(A,bool),fun(multiset(A),multiset(A))),map_fun(fun(A,bool),fun(A,bool),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),id(fun(A,bool)),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_anj(fun(A,bool),fun(fun(A,nat),fun(A,nat)))) ).

% filter_mset_def
tff(fact_6954_subset__mset_OcSUP__union,axiom,
    ! [A: $tType,B: $tType,A6: set(B),F3: fun(B,multiset(A)),B5: set(B)] :
      ( ( A6 != bot_bot(set(B)) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
       => ( ( B5 != bot_bot(set(B)) )
         => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B5)))
           => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),B5))) ) ) ) ) ) ).

% subset_mset.cSUP_union
tff(fact_6955_type__definition__multiset,axiom,
    ! [A: $tType] : type_definition(multiset(A),fun(A,nat),count(A),abs_multiset(A),aa(fun(fun(A,nat),bool),set(fun(A,nat)),collect(fun(A,nat)),aTP_Lamp_akl(fun(A,nat),bool))) ).

% type_definition_multiset
tff(fact_6956_set__mset__sup,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : aa(multiset(A),set(A),set_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(multiset(A),set(A),set_mset(A),A6)),aa(multiset(A),set(A),set_mset(A),B5)) ).

% set_mset_sup
tff(fact_6957_subset__mset_Obdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),G3: fun(B,multiset(A)),A6: set(B)] :
      ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_any(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A6)))
    <=> ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
        & pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A6))) ) ) ).

% subset_mset.bdd_above_image_sup
tff(fact_6958_typedef__rep__transfer,axiom,
    ! [A: $tType,B: $tType,Rep: fun(B,A),Abs: fun(A,B),A6: set(A),T8: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A6)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => pp(aa(fun(B,A),bool,aa(fun(A,A),fun(fun(B,A),bool),bNF_rel_fun(A,B,A,A,T8,fequal(A)),aTP_Lamp_cc(A,A)),Rep)) ) ) ).

% typedef_rep_transfer
tff(fact_6959_type__copy__map__comp0,axiom,
    ! [F2: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: fun(A,B),Abs: fun(B,A),M6: fun(C,D),M12: fun(B,D),M23: fun(C,B),F3: fun(D,F2),G3: fun(E,C)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( M6 = aa(fun(C,B),fun(C,D),comp(B,D,C,M12),M23) )
       => ( aa(fun(E,C),fun(E,F2),comp(C,F2,E,aa(fun(C,D),fun(C,F2),comp(D,F2,C,F3),M6)),G3) = aa(fun(E,A),fun(E,F2),comp(A,F2,E,aa(fun(A,B),fun(A,F2),comp(B,F2,A,aa(fun(B,D),fun(B,F2),comp(D,F2,B,F3),M12)),Rep)),aa(fun(E,C),fun(E,A),comp(C,A,E,aa(fun(C,B),fun(C,A),comp(B,A,C,Abs),M23)),G3)) ) ) ) ).

% type_copy_map_comp0
tff(fact_6960_type__copy__wit,axiom,
    ! [A: $tType,C: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),X: C,S: fun(B,set(C)),Y: B] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( pp(aa(set(C),bool,member(C,X),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),comp(B,set(C),A,S),Rep),aa(B,A,Abs,Y))))
       => pp(aa(set(C),bool,member(C,X),aa(B,set(C),S,Y))) ) ) ).

% type_copy_wit
tff(fact_6961_type__copy__ex__RepI,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),F4: fun(B,bool)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ? [X_12: B] : pp(aa(B,bool,F4,X_12))
      <=> ? [B7: A] : pp(aa(B,bool,F4,aa(A,B,Rep,B7))) ) ) ).

% type_copy_ex_RepI
tff(fact_6962_sup__union__distrib__left,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A),C4: 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)),union_mset(A),A6),B5)),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4)) ).

% sup_union_distrib_left
tff(fact_6963_union__sup__distrib__right,axiom,
    ! [A: $tType,C4: multiset(A),A6: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),A6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),B5)) ).

% union_sup_distrib_right
tff(fact_6964_union__mset__def,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A6),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),A6)) ).

% union_mset_def
tff(fact_6965_union__diff__inter__eq__sup,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),A6),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5) ).

% union_diff_inter_eq_sup
tff(fact_6966_union__diff__sup__eq__inter,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(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)),A6),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5) ).

% union_diff_sup_eq_inter
tff(fact_6967_size__Un__Int,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),A6)),aa(multiset(A),nat,size_size(multiset(A)),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5))),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5))) ).

% size_Un_Int
tff(fact_6968_type__copy__map__id0,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),M6: fun(B,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( M6 = id(B) )
       => ( aa(fun(A,B),fun(A,A),comp(B,A,A,aa(fun(B,B),fun(B,A),comp(B,A,B,Abs),M6)),Rep) = id(A) ) ) ) ).

% type_copy_map_id0
tff(fact_6969_size__Un__disjoint,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A6),B5) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),A6)),aa(multiset(A),nat,size_size(multiset(A)),B5)) ) ) ).

% size_Un_disjoint
tff(fact_6970_subset__mset_Omono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F3: fun(multiset(A),B),A6: multiset(A),B5: multiset(A)] :
          ( pp(aa(fun(multiset(A),B),bool,mono(multiset(A),B,subseteq_mset(A)),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(multiset(A),B,F3,A6)),aa(multiset(A),B,F3,B5))),aa(multiset(A),B,F3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A6),B5)))) ) ) ).

% subset_mset.mono_sup
tff(fact_6971_sum__mset_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: multiset(B),B5: multiset(B),G3: fun(B,A)] :
          ( ( aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),inter_mset(B),A6),B5) = zero_zero(multiset(B)) )
         => ( comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),union_mset(B),A6),B5))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),A6))),comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,G3),B5))) ) ) ) ).

% sum_mset.union_disjoint
tff(fact_6972_subset__mset_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType,A6: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( ( A6 != bot_bot(set(B)) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
       => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A6)))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6))),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),G3),A6))) = aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_any(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),F3),G3)),A6)) ) ) ) ) ).

% subset_mset.SUP_sup_distrib
tff(fact_6973_subset__mset_OcSup__union__distrib,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( ( A6 != bot_bot(set(multiset(A))) )
     => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
       => ( ( B5 != bot_bot(set(multiset(A))) )
         => ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),B5))
           => ( aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),A6)),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),B5)) ) ) ) ) ) ).

% subset_mset.cSup_union_distrib
tff(fact_6974_type__copy__set__map0,axiom,
    ! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F2: $tType,Rep: fun(A,B),Abs: fun(B,A),S: fun(B,set(D)),M6: fun(C,B),F3: fun(E,D),S4: fun(C,set(E)),G3: fun(F2,C)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( aa(fun(C,B),fun(C,set(D)),comp(B,set(D),C,S),M6) = aa(fun(C,set(E)),fun(C,set(D)),comp(set(E),set(D),C,image2(E,D,F3)),S4) )
       => ( aa(fun(F2,A),fun(F2,set(D)),comp(A,set(D),F2,aa(fun(A,B),fun(A,set(D)),comp(B,set(D),A,S),Rep)),aa(fun(F2,C),fun(F2,A),comp(C,A,F2,aa(fun(C,B),fun(C,A),comp(B,A,C,Abs),M6)),G3)) = aa(fun(F2,set(E)),fun(F2,set(D)),comp(set(E),set(D),F2,image2(E,D,F3)),aa(fun(F2,C),fun(F2,set(E)),comp(C,set(E),F2,S4),G3)) ) ) ) ).

% type_copy_set_map0
tff(fact_6975_sum__multiset__singleton,axiom,
    ! [A: $tType,A6: set(A)] : groups7311177749621191930dd_sum(A,multiset(A),aTP_Lamp_ait(A,multiset(A)),A6) = mset_set(A,A6) ).

% sum_multiset_singleton
tff(fact_6976_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : bit_se5824344971392196577ns_xor(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(M,N)) ) ).

% numeral_xor_num
tff(fact_6977_filter__mset__mset__set,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),P2),mset_set(A,A6)) = mset_set(A,aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2))) ) ) ).

% filter_mset_mset_set
tff(fact_6978_msubset__mset__set__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),mset_set(A,A6)),mset_set(A,B5)))
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5)) ) ) ) ).

% msubset_mset_set_iff
tff(fact_6979_subset__imp__msubset__mset__set,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),mset_set(A,A6)),mset_set(A,B5))) ) ) ).

% subset_imp_msubset_mset_set
tff(fact_6980_mset__set__Diff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
       => ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),mset_set(A,A6)),mset_set(A,B5)) ) ) ) ).

% mset_set_Diff
tff(fact_6981_count__mset__set__finite__iff,axiom,
    ! [A: $tType,S: set(A),A3: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( ( pp(aa(set(A),bool,member(A,A3),S))
         => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S)),A3) = one_one(nat) ) )
        & ( ~ pp(aa(set(A),bool,member(A,A3),S))
         => ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S)),A3) = zero_zero(nat) ) ) ) ) ).

% count_mset_set_finite_iff
tff(fact_6982_mset__set__Union,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
         => ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,A6)),mset_set(A,B5)) ) ) ) ) ).

% mset_set_Union
tff(fact_6983_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F4: fun(A,set(B)),F3: fun(B,C),G5: fun(D,set(C)),J4: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => ! [J3: A] :
                ( pp(aa(set(A),bool,member(A,J3),I5))
               => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F4,I3)),aa(A,set(B),F4,J3)))
                  | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F4,J3)),aa(A,set(B),F4,I3))) ) ) )
       => ( filterlim(B,C,F3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_anz(fun(D,set(C)),fun(D,filter(C)),G5)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_agh(fun(A,set(B)),fun(A,filter(B)),F4)),I5)))
        <=> ! [X4: D] :
              ( pp(aa(set(D),bool,member(D,X4),J4))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,member(A,Xa3),I5))
                  & ! [Xb5: B] :
                      ( pp(aa(set(B),bool,member(B,Xb5),aa(A,set(B),F4,Xa3)))
                     => pp(aa(set(C),bool,member(C,aa(B,C,F3,Xb5)),aa(D,set(C),G5,X4))) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_6984_map__rec,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),rec_list(list(A),B,nil(A),aTP_Lamp_aoa(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F3)),Xs) ).

% map_rec
tff(fact_6985_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F23: filter(B),F14: filter(A),F24: filter(B),F15: filter(A)] :
      ( filterlim(A,B,F3,F23,F14)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F23),F24))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F15),F14))
         => filterlim(A,B,F3,F24,F15) ) ) ) ).

% filterlim_mono
tff(fact_6986_rec__list__Nil__imp,axiom,
    ! [A: $tType,B: $tType,F3: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B)))] :
      ( ( F3 = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F3,nil(A)) = F1 ) ) ).

% rec_list_Nil_imp
tff(fact_6987_filterlim__sup,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(B),F14: filter(A),F23: filter(A)] :
      ( filterlim(A,B,F3,F4,F14)
     => ( filterlim(A,B,F3,F4,F23)
       => filterlim(A,B,F3,F4,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F14),F23)) ) ) ).

% filterlim_sup
tff(fact_6988_filterlim__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filterlim(A,A,aTP_Lamp_cc(A,A),F4,F4) ).

% filterlim_ident
tff(fact_6989_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: fun(A,B),F33: filter(B),F23: filter(A),F3: fun(C,A),F14: filter(C)] :
      ( filterlim(A,B,G3,F33,F23)
     => ( filterlim(C,A,F3,F23,F14)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aob(fun(A,B),fun(fun(C,A),fun(C,B)),G3),F3),F33,F14) ) ) ).

% filterlim_compose
tff(fact_6990_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F23: filter(B),F33: filter(B),F14: filter(A)] :
      ( filterlim(A,B,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F33),F14)
    <=> ( filterlim(A,B,F3,F23,F14)
        & filterlim(A,B,F3,F33,F14) ) ) ).

% filterlim_inf
tff(fact_6991_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_sz(fun(nat,A),fun(nat,A),F3),F4,at_top(nat))
    <=> filterlim(nat,A,F3,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6992_rec__list__Cons__imp,axiom,
    ! [B: $tType,A: $tType,F3: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B))),X: A,Xs: list(A)] :
      ( ( F3 = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),F22,X),Xs),aa(list(A),B,F3,Xs)) ) ) ).

% rec_list_Cons_imp
tff(fact_6993_filterlim__INF_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: A,A6: set(A),F3: fun(B,C),F4: filter(C),G5: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,member(A,X),A6))
     => ( filterlim(B,C,F3,F4,aa(A,filter(B),G5,X))
       => filterlim(B,C,F3,F4,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),G5),A6))) ) ) ).

% filterlim_INF'
tff(fact_6994_filterlim__INF,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,B),G5: fun(C,filter(B)),B5: set(C),F4: filter(A)] :
      ( filterlim(A,B,F3,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),G5),B5)),F4)
    <=> ! [X4: C] :
          ( pp(aa(set(C),bool,member(C,X4),B5))
         => filterlim(A,B,F3,aa(C,filter(B),G5,X4),F4) ) ) ).

% filterlim_INF
tff(fact_6995_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G5: filter(B),F4: filter(A),P2: fun(A,bool),G3: fun(A,B)] :
      ( filterlim(A,B,F3,G5,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,aa(fun(A,bool),set(A),collect(A),P2))))
     => ( filterlim(A,B,G3,G5,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,aa(fun(A,bool),set(A),collect(A),aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2)))))
       => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_aoc(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),F3),P2),G3),G5,F4) ) ) ).

% filterlim_If
tff(fact_6996_filterlim__base,axiom,
    ! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J4: set(A),I2: fun(A,C),I5: set(C),F4: fun(C,set(D)),F3: fun(D,E),G5: fun(A,set(E))] :
      ( ! [M5: A,X3: B] :
          ( pp(aa(set(A),bool,member(A,M5),J4))
         => pp(aa(set(C),bool,member(C,aa(A,C,I2,M5)),I5)) )
     => ( ! [M5: A,X3: D] :
            ( pp(aa(set(A),bool,member(A,M5),J4))
           => ( pp(aa(set(D),bool,member(D,X3),aa(C,set(D),F4,aa(A,C,I2,M5))))
             => pp(aa(set(E),bool,member(E,aa(D,E,F3,X3)),aa(A,set(E),G5,M5))) ) )
       => filterlim(D,E,F3,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),aTP_Lamp_aod(fun(A,set(E)),fun(A,filter(E)),G5)),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),image2(C,filter(D),aTP_Lamp_aoe(fun(C,set(D)),fun(C,filter(D)),F4)),I5))) ) ) ).

% filterlim_base
tff(fact_6997_list_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: C,Ga: fun(B,fun(list(B),fun(C,C))),F3: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),C),comp(list(B),C,list(A),rec_list(C,B,G3,Ga)),map(A,B,F3)) = rec_list(C,A,G3,aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_aof(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Ga),F3)) ).

% list.rec_o_map
tff(fact_6998_zipf__zip,axiom,
    ! [A: $tType,B: $tType,L12: list(A),L23: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L23) )
     => ( zipf(A,B,product_prod(A,B),product_Pair(A,B),L12,L23) = zip(A,B,L12,L23) ) ) ).

% zipf_zip
tff(fact_6999_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] :
          ( ( semiring_gcd_Gcd_fin(A,A6) = zero_zero(A) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A)))))
            & pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_7000_zipf_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,fun(B,C))] : zipf(A,B,C,F3,nil(A),nil(B)) = nil(C) ).

% zipf.simps(1)
tff(fact_7001_zipf_Osimps_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,fun(B,C)),A3: A,As: list(A),B2: B,Bs: list(B)] : zipf(A,B,C,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),F3,A3),B2)),zipf(A,B,C,F3,As,Bs)) ).

% zipf.simps(2)
tff(fact_7002_Gcd__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A),B5: set(A)] : semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),semiring_gcd_Gcd_fin(A,A6)),semiring_gcd_Gcd_fin(A,B5)) ) ).

% Gcd_fin.union
tff(fact_7003_Gcd__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B5: set(A),A6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),semiring_gcd_Gcd_fin(A,B5)),semiring_gcd_Gcd_fin(A,A6)) = semiring_gcd_Gcd_fin(A,A6) ) ) ) ).

% Gcd_fin.subset
tff(fact_7004_zipf_Oelims,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: fun(A,fun(B,C)),Xa2: list(A),Xb2: list(B),Y: list(C)] :
      ( ( zipf(A,B,C,X,Xa2,Xb2) = Y )
     => ( ( ( Xa2 = nil(A) )
         => ( ( Xb2 = nil(B) )
           => ( Y != nil(C) ) ) )
       => ( ! [A5: A,As2: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
             => ! [B4: B,Bs2: list(B)] :
                  ( ( Xb2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
                 => ( Y != aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X,A5),B4)),zipf(A,B,C,X,As2,Bs2)) ) ) )
         => ( ( ? [V3: A,Va: list(A)] : Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)
             => ( ( Xb2 = nil(B) )
               => ( Y != undefined(list(C)) ) ) )
           => ~ ( ( Xa2 = nil(A) )
               => ( ? [V3: B,Va: list(B)] : Xb2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va)
                 => ( Y != undefined(list(C)) ) ) ) ) ) ) ) ).

% zipf.elims
tff(fact_7005_set__rec,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_aog(A,fun(list(A),fun(set(A),set(A))))),Xs) ).

% set_rec
tff(fact_7006_zipf_Opelims,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: fun(A,fun(B,C)),Xa2: list(A),Xb2: list(B),Y: list(C)] :
      ( ( zipf(A,B,C,X,Xa2,Xb2) = Y )
     => ( pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),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)),Xa2),Xb2))))
       => ( ( ( Xa2 = nil(A) )
           => ( ( Xb2 = nil(B) )
             => ( ( Y = nil(C) )
               => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),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))))) ) ) )
         => ( ! [A5: A,As2: list(A)] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2) )
               => ! [B4: B,Bs2: list(B)] :
                    ( ( Xb2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2) )
                   => ( ( Y = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X,A5),B4)),zipf(A,B,C,X,As2,Bs2)) )
                     => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs2))))) ) ) )
           => ( ! [V3: A,Va: list(A)] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va) )
                 => ( ( Xb2 = nil(B) )
                   => ( ( Y = undefined(list(C)) )
                     => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va)),nil(B))))) ) ) )
             => ~ ( ( Xa2 = nil(A) )
                 => ! [V3: B,Va: list(B)] :
                      ( ( Xb2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va) )
                     => ( ( Y = undefined(list(C)) )
                       => ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va))))) ) ) ) ) ) ) ) ) ).

% zipf.pelims
tff(fact_7007_filterlim__lessThan__at__top,axiom,
    filterlim(nat,set(nat),set_ord_lessThan(nat),finite5375528669736107172at_top(nat,top_top(set(nat))),at_top(nat)) ).

% filterlim_lessThan_at_top
tff(fact_7008_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A6: set(A)] : finite5375528669736107172at_top(A,A6) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_aoi(set(A),fun(set(A),filter(set(A))),A6)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_aoj(set(A),fun(set(A),bool),A6)))) ).

% finite_subsets_at_top_def
tff(fact_7009_filterlim__atMost__at__top,axiom,
    filterlim(nat,set(nat),set_ord_atMost(nat),finite5375528669736107172at_top(nat,top_top(set(nat))),at_top(nat)) ).

% filterlim_atMost_at_top
tff(fact_7010_curr__surj,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: fun(A,fun(B,C)),A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(fun(A,fun(B,C))),bool,member(fun(A,fun(B,C)),G3),bNF_Wellorder_Func(A,fun(B,C),A6,bNF_Wellorder_Func(B,C,B5,C4))))
     => ? [X3: fun(product_prod(A,B),C)] :
          ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),X3),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),C4)))
          & ( aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A6),X3) = G3 ) ) ) ).

% curr_surj
tff(fact_7011_curr__in,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(product_prod(A,B),C),A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),F3),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),C4)))
     => pp(aa(set(fun(A,fun(B,C))),bool,member(fun(A,fun(B,C)),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A6),F3)),bNF_Wellorder_Func(A,fun(B,C),A6,bNF_Wellorder_Func(B,C,B5,C4)))) ) ).

% curr_in
tff(fact_7012_curr__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,A6: set(A),F3: fun(product_prod(A,B),C),X5: A] :
      ( ( pp(aa(set(A),bool,member(A,X5),A6))
       => ( aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A6),F3),X5) = aa(A,fun(B,C),aTP_Lamp_dr(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3),X5) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X5),A6))
       => ( aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A6),F3),X5) = undefined(fun(B,C)) ) ) ) ).

% curr_def
tff(fact_7013_curr__inj,axiom,
    ! [C: $tType,B: $tType,A: $tType,F1: fun(product_prod(A,B),C),A6: set(A),B5: set(B),C4: set(C),F22: fun(product_prod(A,B),C)] :
      ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),F1),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),C4)))
     => ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),F22),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),C4)))
       => ( ( aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A6),F1) = aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A6),F22) )
        <=> ( F1 = F22 ) ) ) ) ).

% curr_inj
tff(fact_7014_bij__betw__curr,axiom,
    ! [A: $tType,B: $tType,C: $tType,A6: set(A),B5: set(B),C4: set(C)] : bij_betw(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A6),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),C4),bNF_Wellorder_Func(A,fun(B,C),A6,bNF_Wellorder_Func(B,C,B5,C4))) ).

% bij_betw_curr
tff(fact_7015_disjnt__equiv__class,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
      <=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ).

% disjnt_equiv_class
tff(fact_7016_disjnt__self__iff__empty,axiom,
    ! [A: $tType,S: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),S),S))
    <=> ( S = bot_bot(set(A)) ) ) ).

% disjnt_self_iff_empty
tff(fact_7017_disjnt__insert2,axiom,
    ! [A: $tType,Y6: set(A),A3: A,X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)))
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),Y6))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Y6),X6)) ) ) ).

% disjnt_insert2
tff(fact_7018_disjnt__insert1,axiom,
    ! [A: $tType,A3: A,X6: set(A),Y6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)),Y6))
    <=> ( ~ pp(aa(set(A),bool,member(A,A3),Y6))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6)) ) ) ).

% disjnt_insert1
tff(fact_7019_disjnt__Un2,axiom,
    ! [A: $tType,C4: set(A),A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),A6))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),C4),B5)) ) ) ).

% disjnt_Un2
tff(fact_7020_disjnt__Un1,axiom,
    ! [A: $tType,A6: set(A),B5: set(A),C4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)),C4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),C4))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B5),C4)) ) ) ).

% disjnt_Un1
tff(fact_7021_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A6: set(A),B5: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A3),A6,B5)
        <=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),A6) = B5 ) ) ) ).

% bij_betw_add
tff(fact_7022_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C4: set(A),A6: set(B),B5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,C4,aTP_Lamp_aaz(set(B),fun(A,set(B)),A6))),product_Sigma(A,B,C4,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))
    <=> ( ( C4 = bot_bot(set(A)) )
        | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),A6),B5)) ) ) ).

% disjnt_Times1_iff
tff(fact_7023_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A6: set(A),C4: set(B),B5: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))),product_Sigma(A,B,B5,aTP_Lamp_aaz(set(B),fun(A,set(B)),C4))))
    <=> ( ( C4 = bot_bot(set(B)) )
        | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B5)) ) ) ).

% disjnt_Times2_iff
tff(fact_7024_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),A13: set(B),B5: set(A),B13: set(B)] :
      ( bij_betw(A,B,F3,A6,A13)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
       => ( ( aa(set(A),set(B),image2(A,B,F3),B5) = B13 )
         => bij_betw(A,B,F3,B5,B13) ) ) ) ).

% bij_betw_subset
tff(fact_7025_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F5: fun(B,A),F3: fun(A,B),A13: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => ( aa(B,A,F5,aa(A,B,F3,X3)) = X3 ) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),A13))
           => ( aa(A,B,F3,aa(B,A,F5,X3)) = X3 ) )
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),A13))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F5),A13)),A6))
           => bij_betw(A,B,F3,A6,A13) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_7026_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),B5: set(B),G3: fun(B,A)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),B5))
       => ( inj_on(B,A,G3,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G3),B5)),A6))
           => ? [H2: fun(A,B)] : bij_betw(A,B,H2,A6,B5) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_7027_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F5: fun(A,B),A13: set(A),A17: set(B),F3: fun(C,A),A6: set(C)] :
      ( bij_betw(A,B,F5,A13,A17)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F3),A6)),A13))
       => ( bij_betw(C,A,F3,A6,A13)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F5),F3),A6,A17) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_7028_disjnt__subset2,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),Z5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z5),Y6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Z5)) ) ) ).

% disjnt_subset2
tff(fact_7029_disjnt__subset1,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),Z5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z5),X6))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),Z5),Y6)) ) ) ).

% disjnt_subset1
tff(fact_7030_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),C4: set(A),B5: set(B),D5: set(B)] :
      ( bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),C4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D5))
     => ( bij_betw(A,B,F3,C4,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C4) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B5),D5) = bot_bot(set(B)) )
           => bij_betw(A,B,F3,A6,B5) ) ) ) ) ).

% bij_betw_partition
tff(fact_7031_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),B5: set(B),C4: set(A),D5: set(B)] :
      ( bij_betw(A,B,F3,A6,B5)
     => ( bij_betw(A,B,F3,C4,D5)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B5),D5) = bot_bot(set(B)) )
         => bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),C4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D5)) ) ) ) ).

% bij_betw_combine
tff(fact_7032_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A6: set(A),F3: fun(A,B),A13: set(B)] :
      ( ~ pp(aa(set(A),bool,member(A,B2),A6))
     => ( ~ pp(aa(set(B),bool,member(B,aa(A,B,F3,B2)),A13))
       => ( bij_betw(A,B,F3,A6,A13)
        <=> bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A13),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_7033_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A6: set(A),F3: fun(A,B),A13: set(B)] :
      ( ~ pp(aa(set(A),bool,member(A,B2),A6))
     => ( ~ pp(aa(set(B),bool,member(B,aa(A,B,F3,B2)),A13))
       => ( bij_betw(A,B,F3,A6,A13)
         => bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A13),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_7034_disjnt__insert,axiom,
    ! [A: $tType,X: A,N7: set(A),M6: set(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),N7))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),M6),N7))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),M6)),N7)) ) ) ).

% disjnt_insert
tff(fact_7035_prod_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [H: fun(B,C),S: set(B),T8: set(C),G3: fun(C,A)] :
          ( bij_betw(B,C,H,S,T8)
         => ( groups7121269368397514597t_prod(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_aok(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3),S) = groups7121269368397514597t_prod(C,A,G3,T8) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_7036_disjnt__iff,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B5))
    <=> ! [X4: A] :
          ~ ( pp(aa(set(A),bool,member(A,X4),A6))
            & pp(aa(set(A),bool,member(A,X4),B5)) ) ) ).

% disjnt_iff
tff(fact_7037_disjnt__sym,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),B5),A6)) ) ).

% disjnt_sym
tff(fact_7038_bij__betwI_H,axiom,
    ! [A: $tType,B: $tType,X6: set(A),F3: fun(A,B),Y6: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),X6))
         => ! [Y3: A] :
              ( pp(aa(set(A),bool,member(A,Y3),X6))
             => ( ( aa(A,B,F3,X3) = aa(A,B,F3,Y3) )
              <=> ( X3 = Y3 ) ) ) )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),X6))
           => pp(aa(set(B),bool,member(B,aa(A,B,F3,X3)),Y6)) )
       => ( ! [Y3: B] :
              ( pp(aa(set(B),bool,member(B,Y3),Y6))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,member(A,X5),X6))
                  & ( Y3 = aa(A,B,F3,X5) ) ) )
         => bij_betw(A,B,F3,X6,Y6) ) ) ) ).

% bij_betwI'
tff(fact_7039_bij__betw__funpow,axiom,
    ! [A: $tType,F3: fun(A,A),S: set(A),N: nat] :
      ( bij_betw(A,A,F3,S,S)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),S,S) ) ).

% bij_betw_funpow
tff(fact_7040_bij__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_7041_sum_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [H: fun(B,C),S: set(B),T8: set(C),G3: fun(C,A)] :
          ( bij_betw(B,C,H,S,T8)
         => ( groups7311177749621191930dd_sum(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_aol(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3),S) = groups7311177749621191930dd_sum(C,A,G3,T8) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_7042_disjnt__def,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B5))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) ) ) ).

% disjnt_def
tff(fact_7043_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A6: set(A),C4: fun(A,set(B)),B5: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A6,C4)),product_Sigma(A,B,B5,C4)))
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))
           => ( aa(A,set(B),C4,X4) = bot_bot(set(B)) ) )
        | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B5)) ) ) ).

% disjnt_Sigma_iff
tff(fact_7044_disjnt__empty2,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),bot_bot(set(A)))) ).

% disjnt_empty2
tff(fact_7045_disjnt__empty1,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),bot_bot(set(A))),A6)) ).

% disjnt_empty1
tff(fact_7046_disjoint__UN__iff,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: fun(B,set(A)),I5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),I5))))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),aa(B,set(A),B5,X4))) ) ) ).

% disjoint_UN_iff
tff(fact_7047_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),C4: set(B),G3: fun(A,B),B5: set(A),D5: set(B)] :
      ( bij_betw(A,B,F3,A6,C4)
     => ( bij_betw(A,B,G3,B5,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C4),D5) = bot_bot(set(B)) )
           => bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_ahq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F3),A6),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C4),D5)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7048_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),F3: fun(B,C),A13: fun(A,set(C))] :
      ( ! [I3: A,J3: A] :
          ( pp(aa(set(A),bool,member(A,I3),I5))
         => ( pp(aa(set(A),bool,member(A,J3),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,I3)),aa(A,set(B),A6,J3)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,J3)),aa(A,set(B),A6,I3))) ) ) )
     => ( ! [I3: A] :
            ( pp(aa(set(A),bool,member(A,I3),I5))
           => bij_betw(B,C,F3,aa(A,set(B),A6,I3),aa(A,set(C),A13,I3)) )
       => bij_betw(B,C,F3,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A6),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A13),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_7049_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A6: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_7050_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y6: set(A),X6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),Y6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),Y6)),X3)) )
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y6)) ) ) ) ).

% disjnt_ge_max
tff(fact_7051_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(B),A6: set(A)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F3),B5)),A6))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F3),A6))) ) ) ).

% vimage_subset_eq
tff(fact_7052_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S4: set(B),T10: set(C),H: fun(B,C),S: set(B),T8: set(C),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S4))
         => ( pp(aa(set(C),bool,finite_finite2(C),T10))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T8),T10))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,member(B,A5),S4))
                   => ( aa(C,A,G3,aa(B,C,H,A5)) = zero_zero(A) ) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,member(C,B4),T10))
                     => ( aa(C,A,G3,B4) = zero_zero(A) ) )
                 => ( groups7311177749621191930dd_sum(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_aol(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3),S) = groups7311177749621191930dd_sum(C,A,G3,T8) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_7053_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S4: set(B),T10: set(C),H: fun(B,C),S: set(B),T8: set(C),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S4))
         => ( pp(aa(set(C),bool,finite_finite2(C),T10))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S),S4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T8),T10))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,member(B,A5),S4))
                   => ( aa(C,A,G3,aa(B,C,H,A5)) = one_one(A) ) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,member(C,B4),T10))
                     => ( aa(C,A,G3,B4) = one_one(A) ) )
                 => ( groups7121269368397514597t_prod(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_aok(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3),S) = groups7121269368397514597t_prod(C,A,G3,T8) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_7054_bij__image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),B5: fun(C,set(A)),A6: set(C)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B5),A6))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_ahr(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F3),B5)),A6)) ) ) ).

% bij_image_INT
tff(fact_7055_card__Un__disjnt,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B5))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B5))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_7056_sum__card__image,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pairwise(A,aTP_Lamp_aom(fun(A,set(B)),fun(A,fun(A,bool)),F3),A6)
       => ( groups7311177749621191930dd_sum(set(B),nat,finite_card(B),aa(set(A),set(set(B)),image2(A,set(B),F3),A6)) = groups7311177749621191930dd_sum(A,nat,aTP_Lamp_yb(fun(A,set(B)),fun(A,nat),F3),A6) ) ) ) ).

% sum_card_image
tff(fact_7057_effect__makeE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,F3: fun(nat,A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: array(A),N2: nat] :
          ( heap_Time_effect(array(A),array_make(A,N,F3),H,H5,R3,N2)
         => ~ ( ( R3 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),N)),H)) )
             => ( ( H5 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),N)),H)) )
               => ( ( array_get(A,H5,R3) = aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),N)) )
                 => ( array_present(A,H5,R3)
                   => ( ~ array_present(A,H,R3)
                     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N2 ) ) ) ) ) ) ) ) ).

% effect_makeE
tff(fact_7058_bij__betw__Suc,axiom,
    ! [M6: set(nat),N7: set(nat)] :
      ( bij_betw(nat,nat,suc,M6,N7)
    <=> ( aa(set(nat),set(nat),image2(nat,nat,suc),M6) = N7 ) ) ).

% bij_betw_Suc
tff(fact_7059_bij__swap,axiom,
    ! [A: $tType,B: $tType] : bij_betw(product_prod(A,B),product_prod(B,A),product_swap(A,B),top_top(set(product_prod(A,B))),top_top(set(product_prod(B,A)))) ).

% bij_swap
tff(fact_7060_pairwise__empty,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool))] : pairwise(A,P2,bot_bot(set(A))) ).

% pairwise_empty
tff(fact_7061_pairwise__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),S: set(A)] :
      ( pairwise(A,R2,S)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),S))
             => ( ( X4 != Xa3 )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),Xa3)) ) ) ) ) ).

% pairwise_def
tff(fact_7062_pairwiseI,axiom,
    ! [A: $tType,S: set(A),R2: fun(A,fun(A,bool))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(A),bool,member(A,X3),S))
         => ( pp(aa(set(A),bool,member(A,Y3),S))
           => ( ( X3 != Y3 )
             => pp(aa(A,bool,aa(A,fun(A,bool),R2,X3),Y3)) ) ) )
     => pairwise(A,R2,S) ) ).

% pairwiseI
tff(fact_7063_pairwiseD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),S: set(A),X: A,Y: A] :
      ( pairwise(A,R2,S)
     => ( pp(aa(set(A),bool,member(A,X),S))
       => ( pp(aa(set(A),bool,member(A,Y),S))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),R2,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_7064_pairwise__trivial,axiom,
    ! [A: $tType,I5: set(A)] : pairwise(A,aTP_Lamp_qp(A,fun(A,bool)),I5) ).

% pairwise_trivial
tff(fact_7065_pairwise__insert,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),X: A,S2: set(A)] :
      ( pairwise(A,R3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S2))
    <=> ( ! [Y4: A] :
            ( ( pp(aa(set(A),bool,member(A,Y4),S2))
              & ( Y4 != X ) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R3,X),Y4))
              & pp(aa(A,bool,aa(A,fun(A,bool),R3,Y4),X)) ) )
        & pairwise(A,R3,S2) ) ) ).

% pairwise_insert
tff(fact_7066_pairwise__mono,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),A6: set(A),Q2: fun(A,fun(A,bool)),B5: set(A)] :
      ( pairwise(A,P2,A6)
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Y3))
           => pp(aa(A,bool,aa(A,fun(A,bool),Q2,X3),Y3)) )
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6))
         => pairwise(A,Q2,B5) ) ) ) ).

% pairwise_mono
tff(fact_7067_pairwise__subset,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),S: set(A),T8: set(A)] :
      ( pairwise(A,P2,S)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T8),S))
       => pairwise(A,P2,T8) ) ) ).

% pairwise_subset
tff(fact_7068_pairwise__imageI,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B),P2: fun(B,fun(B,bool))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A6))
         => ( pp(aa(set(A),bool,member(A,Y3),A6))
           => ( ( X3 != Y3 )
             => ( ( aa(A,B,F3,X3) != aa(A,B,F3,Y3) )
               => pp(aa(B,bool,aa(B,fun(B,bool),P2,aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) ) )
     => pairwise(B,P2,aa(set(A),set(B),image2(A,B,F3),A6)) ) ).

% pairwise_imageI
tff(fact_7069_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(A,bool)),F3: fun(B,A),S2: set(B)] :
      ( pairwise(A,R3,aa(set(B),set(A),image2(B,A,F3),S2))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_aon(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R3),F3),S2) ) ).

% pairwise_image
tff(fact_7070_pairwise__singleton,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),A6: A] : pairwise(A,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A6),bot_bot(set(A)))) ).

% pairwise_singleton
tff(fact_7071_pairwise__alt,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool)),S: set(A)] :
      ( pairwise(A,R2,S)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))))))
             => pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),Xa3)) ) ) ) ).

% pairwise_alt
tff(fact_7072_disjoint__image__subset,axiom,
    ! [A: $tType,A18: set(set(A)),F3: fun(set(A),set(A))] :
      ( pairwise(set(A),disjnt(A),A18)
     => ( ! [X10: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X10),A18))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),F3,X10)),X10)) )
       => pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),image2(set(A),set(A),F3),A18)) ) ) ).

% disjoint_image_subset
tff(fact_7073_effect__of__listE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: array(A),N2: nat] :
          ( heap_Time_effect(array(A),array_of_list(A,Xs),H,H5,R3,N2)
         => ~ ( ( R3 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H)) )
             => ( ( H5 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H)) )
               => ( ( array_get(A,H5,R3) = Xs )
                 => ( array_present(A,H5,R3)
                   => ( ~ array_present(A,H,R3)
                     => ( N2 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ) ) ) ) ) ) ).

% effect_of_listE
tff(fact_7074_effect__newE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,X: A,H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: array(A),N2: nat] :
          ( heap_Time_effect(array(A),array_new(A,N,X),H,H5,R3,N2)
         => ~ ( ( R3 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,replicate(A,N,X),H)) )
             => ( ( H5 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,replicate(A,N,X),H)) )
               => ( ( array_get(A,H5,R3) = replicate(A,N,X) )
                 => ( array_present(A,H5,R3)
                   => ( ~ array_present(A,H,R3)
                     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N2 ) ) ) ) ) ) ) ) ).

% effect_newE
tff(fact_7075_execute__nth_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,array_nth(A,A3,I2)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,H,A3)),I2)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ) ) ).

% execute_nth(1)
tff(fact_7076_Array__Time_Onth__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),I2: nat] : array_nth(A,A3,I2) = heap_Time_guard(A,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_aoo(array(A),fun(nat,fun(heap_ext(product_unit),bool)),A3),I2),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aop(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),A3),I2)) ) ).

% Array_Time.nth_def
tff(fact_7077_len__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A)] : array_len(A,A3) = heap_Time_tap(nat,aTP_Lamp_aoq(array(A),fun(heap_ext(product_unit),nat),A3)) ) ).

% len_def
tff(fact_7078_success__updI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => heap_Time_success(array(A),array_upd(A,I2,X,A3),H) ) ) ).

% success_updI
tff(fact_7079_success__nthI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => heap_Time_success(A,array_nth(A,A3,I2),H) ) ) ).

% success_nthI
tff(fact_7080_execute__len,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(nat,array_len(A,A3)),H) = aa(product_prod(nat,product_prod(heap_ext(product_unit),nat)),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),some(product_prod(nat,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat))),product_Pair(nat,product_prod(heap_ext(product_unit),nat)),array_length(A,H,A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ).

% execute_len
tff(fact_7081_execute__upd_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H: heap_ext(product_unit),A3: array(A),I2: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),array_length(A,H,A3)),I2))
         => ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_upd(A,I2,X,A3)),H) = none(product_prod(array(A),product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_upd(2)
tff(fact_7082_effect__freezeI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H5: heap_ext(product_unit),H: heap_ext(product_unit),R3: list(A),A3: array(A),N: nat] :
          ( ( H5 = H )
         => ( ( R3 = array_get(A,H,A3) )
           => ( ( N = array_length(A,H,A3) )
             => heap_Time_effect(list(A),array_freeze(A,A3),H,H5,R3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ) ) ).

% effect_freezeI
tff(fact_7083_effect__freezeE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: list(A),N: nat] :
          ( heap_Time_effect(list(A),array_freeze(A,A3),H,H5,R3,N)
         => ~ ( ( H5 = H )
             => ( ( R3 = array_get(A,H,A3) )
               => ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),array_length(A,H,A3)),one_one(nat)) ) ) ) ) ) ).

% effect_freezeE
tff(fact_7084_execute__nth_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H: heap_ext(product_unit),A3: array(A),I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),array_length(A,H,A3)),I2))
         => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,array_nth(A,A3,I2)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_nth(2)
tff(fact_7085_execute__freeze,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(list(A),array_freeze(A,A3)),H) = aa(product_prod(list(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,H,A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),array_length(A,H,A3))))) ) ).

% execute_freeze
tff(fact_7086_freeze__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A)] : array_freeze(A,A3) = heap_Time_heap(list(A),aTP_Lamp_aor(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),A3)) ) ).

% freeze_def
tff(fact_7087_effect__nthE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),I2: nat,H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
          ( heap_Time_effect(A,array_nth(A,A3,I2),H,H5,R3,N)
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
             => ( ( R3 = aa(nat,A,nth(A,array_get(A,H,A3)),I2) )
               => ( ( H5 = H )
                 => ( N != one_one(nat) ) ) ) ) ) ) ).

% effect_nthE
tff(fact_7088_effect__nthI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),H5: heap_ext(product_unit),R3: A,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( ( H5 = H )
           => ( ( R3 = aa(nat,A,nth(A,array_get(A,H,A3)),I2) )
             => ( ( N = one_one(nat) )
               => heap_Time_effect(A,array_nth(A,A3,I2),H,H5,R3,N) ) ) ) ) ) ).

% effect_nthI
tff(fact_7089_execute__upd_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_upd(A,I2,X,A3)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),A3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I2,X,H)),one_one(nat)))) ) ) ) ).

% execute_upd(1)
tff(fact_7090_execute__swap_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H: heap_ext(product_unit),A3: array(A),I2: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),array_length(A,H,A3)),I2))
         => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,array_swap(A,I2,X,A3)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_swap(2)
tff(fact_7091_effect__swapE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,X: A,A3: array(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: A,N: nat] :
          ( heap_Time_effect(A,array_swap(A,I2,X,A3),H,H5,R3,N)
         => ~ ( ( R3 = aa(nat,A,nth(A,array_get(A,H,A3)),I2) )
             => ( ( H5 = array_update(A,A3,I2,X,H) )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
                 => ( N != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)) ) ) ) ) ) ) ).

% effect_swapE
tff(fact_7092_effect__swapI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),H5: heap_ext(product_unit),X: A,R3: A,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( ( H5 = array_update(A,A3,I2,X,H) )
           => ( ( R3 = aa(nat,A,nth(A,array_get(A,H,A3)),I2) )
             => ( ( N = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)) )
               => heap_Time_effect(A,array_swap(A,I2,X,A3),H,H5,R3,N) ) ) ) ) ) ).

% effect_swapI
tff(fact_7093_effect__updE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,V2: A,A3: array(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: array(A),N: nat] :
          ( heap_Time_effect(array(A),array_upd(A,I2,V2,A3),H,H5,R3,N)
         => ~ ( ( R3 = A3 )
             => ( ( H5 = array_update(A,A3,I2,V2,H) )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
                 => ( N != one_one(nat) ) ) ) ) ) ) ).

% effect_updE
tff(fact_7094_effect__updI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),H5: heap_ext(product_unit),V2: A,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( ( H5 = array_update(A,A3,I2,V2,H) )
           => ( ( N = one_one(nat) )
             => heap_Time_effect(array(A),array_upd(A,I2,V2,A3),H,H5,A3,N) ) ) ) ) ).

% effect_updI
tff(fact_7095_success__swapI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => heap_Time_success(A,array_swap(A,I2,X,A3),H) ) ) ).

% success_swapI
tff(fact_7096_swap__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,X: A,A3: array(A)] : array_swap(A,I2,X,A3) = heap_Time_guard(A,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_aos(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I2),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aot(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),I2),X),A3)) ) ).

% swap_def
tff(fact_7097_execute__swap_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,array_swap(A,I2,X,A3)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,H,A3)),I2)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I2,X,H)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ) ) ).

% execute_swap(1)
tff(fact_7098_upd__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,X: A,A3: array(A)] : array_upd(A,I2,X,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_aos(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I2),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aou(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I2),X),A3)) ) ).

% upd_def
tff(fact_7099_execute__map__entry_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_map_entry(A,I2,F3,A3)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),A3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I2,aa(A,A,F3,aa(nat,A,nth(A,array_get(A,H,A3)),I2)),H)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ) ) ).

% execute_map_entry(1)
tff(fact_7100_map__entry__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,F3: fun(A,A),A3: array(A)] : array_map_entry(A,I2,F3,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_aos(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I2),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aov(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I2),F3),A3)) ) ).

% map_entry_def
tff(fact_7101_success__map__entryI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => heap_Time_success(array(A),array_map_entry(A,I2,F3,A3),H) ) ) ).

% success_map_entryI
tff(fact_7102_execute__map__entry_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H: heap_ext(product_unit),A3: array(A),I2: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),array_length(A,H,A3)),I2))
         => ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_map_entry(A,I2,F3,A3)),H) = none(product_prod(array(A),product_prod(heap_ext(product_unit),nat))) ) ) ) ).

% execute_map_entry(2)
tff(fact_7103_effect__map__entryI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,H: heap_ext(product_unit),A3: array(A),H5: heap_ext(product_unit),F3: fun(A,A),R3: array(A),N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
         => ( ( H5 = array_update(A,A3,I2,aa(A,A,F3,aa(nat,A,nth(A,array_get(A,H,A3)),I2)),H) )
           => ( ( R3 = A3 )
             => ( ( N = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)) )
               => heap_Time_effect(array(A),array_map_entry(A,I2,F3,A3),H,H5,R3,N) ) ) ) ) ) ).

% effect_map_entryI
tff(fact_7104_effect__map__entryE,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I2: nat,F3: fun(A,A),A3: array(A),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: array(A),N: nat] :
          ( heap_Time_effect(array(A),array_map_entry(A,I2,F3,A3),H,H5,R3,N)
         => ~ ( ( R3 = A3 )
             => ( ( H5 = array_update(A,A3,I2,aa(A,A,F3,aa(nat,A,nth(A,array_get(A,H,A3)),I2)),H) )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),array_length(A,H,A3)))
                 => ( N != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)) ) ) ) ) ) ) ).

% effect_map_entryE
tff(fact_7105_init__seg__of__def,axiom,
    ! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),bool),aTP_Lamp_aow(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).

% init_seg_of_def
tff(fact_7106_sqr_Osimps_I3_J,axiom,
    ! [N: num] : sqr(aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(N)),N))) ).

% sqr.simps(3)
tff(fact_7107_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),A3: A] :
          ( inj_on(A,B,F3,aa(fun(A,bool),set(A),collect(A),P2))
         => ( pp(aa(A,bool,P2,A3))
           => ( ! [Y3: A] :
                  ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),aa(A,B,F3,Y3))) )
             => ( lattices_ord_arg_min(A,B,F3,P2) = A3 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_7108_minus__coset__filter,axiom,
    ! [A: $tType,A6: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),coset(A,Xs)) = aa(list(A),set(A),set2(A),filter2(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6),Xs)) ).

% minus_coset_filter
tff(fact_7109_arg__min__equality,axiom,
    ! [A: $tType,C: $tType] :
      ( order(A)
     => ! [P2: fun(C,bool),K2: C,F3: fun(C,A)] :
          ( pp(aa(C,bool,P2,K2))
         => ( ! [X3: C] :
                ( pp(aa(C,bool,P2,X3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,K2)),aa(C,A,F3,X3))) )
           => ( aa(C,A,F3,lattices_ord_arg_min(C,A,F3,P2)) = aa(C,A,F3,K2) ) ) ) ) ).

% arg_min_equality
tff(fact_7110_arg__min__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P2,K2))
     => ( pp(aa(A,bool,P2,lattices_ord_arg_min(A,nat,M,P2)))
        & ! [Y5: A] :
            ( pp(aa(A,bool,P2,Y5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P2))),aa(A,nat,M,Y5))) ) ) ) ).

% arg_min_nat_lemma
tff(fact_7111_arg__min__nat__le,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P2,X))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P2))),aa(A,nat,M,X))) ) ).

% arg_min_nat_le
tff(fact_7112_arg__minI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P2: fun(A,bool),X: A,F3: fun(A,B),Q2: fun(A,bool)] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y3)),aa(A,B,F3,X))) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P2,X3))
                 => ( ! [Y5: A] :
                        ( pp(aa(A,bool,P2,Y5))
                       => ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y5)),aa(A,B,F3,X3))) )
                   => pp(aa(A,bool,Q2,X3)) ) )
             => pp(aa(A,bool,Q2,lattices_ord_arg_min(A,B,F3,P2))) ) ) ) ) ).

% arg_minI
tff(fact_7113_subset__code_I2_J,axiom,
    ! [B: $tType,A6: set(B),Ys2: list(B)] :
      ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),coset(B,Ys2)))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),aa(list(B),set(B),set2(B),Ys2)))
         => ~ pp(aa(set(B),bool,member(B,X4),A6)) ) ) ).

% subset_code(2)
tff(fact_7114_union__coset__filter,axiom,
    ! [A: $tType,Xs: list(A),A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A6) = coset(A,filter2(A,aTP_Lamp_dh(set(A),fun(A,bool),A6),Xs)) ).

% union_coset_filter
tff(fact_7115_arg__min__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: fun(B,A),S: set(B)] : lattic7623131987881927897min_on(B,A,F3,S) = lattices_ord_arg_min(B,A,F3,aTP_Lamp_be(set(B),fun(B,bool),S)) ) ).

% arg_min_on_def
tff(fact_7116_subset__code_I3_J,axiom,
    ! [C: $tType] : ~ pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),coset(C,nil(C))),aa(list(C),set(C),set2(C),nil(C)))) ).

% subset_code(3)
tff(fact_7117_listrel__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),product_Sigma(list(A),list(A),lists(A,A6),aTP_Lamp_aox(set(A),fun(list(A),set(list(A))),A6)))) ) ).

% listrel_subset
tff(fact_7118_multiset_Odomain,axiom,
    ! [C: $tType,B: $tType,T8: fun(C,fun(B,bool)),X5: fun(C,nat)] :
      ( pp(aa(fun(C,nat),bool,aa(fun(fun(C,nat),fun(multiset(B),bool)),fun(fun(C,nat),bool),domainp(fun(C,nat),multiset(B)),pcr_multiset(C,B,T8)),X5))
    <=> ? [Y4: fun(B,nat)] :
          ( pp(aa(fun(B,nat),bool,aa(fun(C,nat),fun(fun(B,nat),bool),bNF_rel_fun(C,B,nat,nat,T8,fequal(nat)),X5),Y4))
          & pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_aoy(fun(B,nat),fun(B,bool),Y4)))) ) ) ).

% multiset.domain
tff(fact_7119_arg__min__natI,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P2,K2))
     => pp(aa(A,bool,P2,lattices_ord_arg_min(A,nat,M,P2))) ) ).

% arg_min_natI
tff(fact_7120_lists__eq__set,axiom,
    ! [A: $tType,A6: set(A)] : lists(A,A6) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_aoz(set(A),fun(list(A),bool),A6)) ).

% lists_eq_set
tff(fact_7121_lists__image__witness,axiom,
    ! [A: $tType,B: $tType,X: list(A),F3: fun(B,A),Q2: set(B)] :
      ( pp(aa(set(list(A)),bool,member(list(A),X),lists(A,aa(set(B),set(A),image2(B,A,F3),Q2))))
     => ~ ! [Xo2: list(B)] :
            ( pp(aa(set(list(B)),bool,member(list(B),Xo2),lists(B,Q2)))
           => ( X != aa(list(B),list(A),map(B,A,F3),Xo2) ) ) ) ).

% lists_image_witness
tff(fact_7122_lists__mono,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),lists(A,A6)),lists(A,B5))) ) ).

% lists_mono
tff(fact_7123_lists__of__len__fin1,axiom,
    ! [A: $tType,P2: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),P2))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P2)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_apa(nat,fun(list(A),bool),N))))) ) ).

% lists_of_len_fin1
tff(fact_7124_lists__of__len__fin2,axiom,
    ! [A: $tType,P2: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),P2))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,P2)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_apb(nat,fun(list(A),bool),N))))) ) ).

% lists_of_len_fin2
tff(fact_7125_multiset_Odomain__eq,axiom,
    ! [A: $tType,X5: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aa(fun(fun(A,nat),fun(multiset(A),bool)),fun(fun(A,nat),bool),domainp(fun(A,nat),multiset(A)),pcr_multiset(A,A,fequal(A))),X5))
    <=> pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_kx(fun(A,nat),fun(A,bool),X5)))) ) ).

% multiset.domain_eq
tff(fact_7126_multiset_Odomain__par__left__total,axiom,
    ! [B: $tType,C: $tType,T8: fun(C,fun(B,bool)),P4: fun(fun(C,nat),bool)] :
      ( left_total(fun(C,nat),fun(B,nat),bNF_rel_fun(C,B,nat,nat,T8,fequal(nat)))
     => ( pp(aa(fun(fun(B,nat),bool),bool,aa(fun(fun(C,nat),bool),fun(fun(fun(B,nat),bool),bool),bNF_rel_fun(fun(C,nat),fun(B,nat),bool,bool,bNF_rel_fun(C,B,nat,nat,T8,fequal(nat)),fequal(bool)),P4),aTP_Lamp_apc(fun(B,nat),bool)))
       => ( aa(fun(fun(C,nat),fun(multiset(B),bool)),fun(fun(C,nat),bool),domainp(fun(C,nat),multiset(B)),pcr_multiset(C,B,T8)) = P4 ) ) ) ).

% multiset.domain_par_left_total
tff(fact_7127_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => boolea2506097494486148201lgebra(A,inf_inf(A),sup_sup(A),uminus_uminus(A),bot_bot(A),top_top(A)) ) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_7128_multiset_Odomain__par,axiom,
    ! [B: $tType,C: $tType,T8: fun(C,fun(B,bool)),DT: fun(C,bool),DS: fun(nat,bool),P23: fun(fun(C,nat),bool)] :
      ( ( aa(fun(C,fun(B,bool)),fun(C,bool),domainp(C,B),T8) = DT )
     => ( ( aa(fun(nat,fun(nat,bool)),fun(nat,bool),domainp(nat,nat),fequal(nat)) = DS )
       => ( left_unique(C,B,T8)
         => ( pp(aa(fun(fun(B,nat),bool),bool,aa(fun(fun(C,nat),bool),fun(fun(fun(B,nat),bool),bool),bNF_rel_fun(fun(C,nat),fun(B,nat),bool,bool,bNF_rel_fun(C,B,nat,nat,T8,fequal(nat)),fequal(bool)),P23),aTP_Lamp_apc(fun(B,nat),bool)))
           => ( aa(fun(fun(C,nat),fun(multiset(B),bool)),fun(fun(C,nat),bool),domainp(fun(C,nat),multiset(B)),pcr_multiset(C,B,T8)) = aa(fun(fun(C,nat),bool),fun(fun(C,nat),bool),aa(fun(fun(C,nat),bool),fun(fun(fun(C,nat),bool),fun(fun(C,nat),bool)),inf_inf(fun(fun(C,nat),bool)),aa(fun(nat,bool),fun(fun(C,nat),bool),basic_pred_fun(C,nat,DT),DS)),P23) ) ) ) ) ) ).

% multiset.domain_par
tff(fact_7129_mono__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType] :
      ( ( order(B)
        & order(D)
        & order(C)
        & order(A) )
     => ! [A6: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] :
          ( bi_total(A,B,A6)
         => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),bool),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A6,bNF_rel_fun(A,B,bool,bool,A6,fequal(bool))),ord_less_eq(A)),ord_less_eq(B)))
           => ( pp(aa(fun(D,fun(D,bool)),bool,aa(fun(C,fun(C,bool)),fun(fun(D,fun(D,bool)),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B5,bNF_rel_fun(C,D,bool,bool,B5,fequal(bool))),ord_less_eq(C)),ord_less_eq(D)))
             => pp(aa(fun(fun(B,D),bool),bool,aa(fun(fun(A,C),bool),fun(fun(fun(B,D),bool),bool),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A6,B5),fequal(bool)),order_mono(A,C)),order_mono(B,D))) ) ) ) ) ).

% mono_transfer
tff(fact_7130_fun_Opred__True,axiom,
    ! [A: $tType,D: $tType,X5: fun(D,A)] : pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),aTP_Lamp_av(A,bool)),X5)) ).

% fun.pred_True
tff(fact_7131_pred__fun__def,axiom,
    ! [B: $tType,A: $tType,A6: fun(A,bool),B5: fun(B,bool),X5: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(fun(B,bool),fun(fun(A,B),bool),basic_pred_fun(A,B,A6),B5),X5))
    <=> ! [Xa3: A] :
          ( pp(aa(A,bool,A6,Xa3))
         => pp(aa(B,bool,B5,aa(A,B,X5,Xa3))) ) ) ).

% pred_fun_def
tff(fact_7132_fun_Opred__mono,axiom,
    ! [D: $tType,A: $tType,P2: fun(A,bool),Pa: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Pa))
     => pp(aa(fun(fun(D,A),bool),bool,aa(fun(fun(D,A),bool),fun(fun(fun(D,A),bool),bool),ord_less_eq(fun(fun(D,A),bool)),aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),P2)),aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),Pa))) ) ).

% fun.pred_mono
tff(fact_7133_typedef__left__unique,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A6: set(A),T8: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A6)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => left_unique(A,B,T8) ) ) ).

% typedef_left_unique
tff(fact_7134_fun_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,D: $tType,X: fun(D,A),Ya: fun(D,A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),aa(fun(A,B),fun(A,bool),aTP_Lamp_ape(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),G3)),Ya))
       => ( aa(fun(D,A),fun(D,B),comp(A,B,D,F3),X) = aa(fun(D,A),fun(D,B),comp(A,B,D,G3),Ya) ) ) ) ).

% fun.map_cong_pred
tff(fact_7135_pred__fun__True__id,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: fun(B,bool),F3: fun(C,B)] :
      ( nO_MATCH(fun(A,A),fun(B,bool),id(A),P3)
     => ( pp(aa(fun(C,B),bool,aa(fun(B,bool),fun(fun(C,B),bool),basic_pred_fun(C,B,aTP_Lamp_apf(C,bool)),P3),F3))
      <=> pp(aa(fun(C,bool),bool,aa(fun(bool,bool),fun(fun(C,bool),bool),basic_pred_fun(C,bool,aTP_Lamp_apf(C,bool)),id(bool)),aa(fun(C,B),fun(C,bool),comp(B,bool,C,P3),F3))) ) ) ).

% pred_fun_True_id
tff(fact_7136_fun_Opred__cong,axiom,
    ! [A: $tType,D: $tType,X: fun(D,A),Ya: fun(D,A),P2: fun(A,bool),Pa: fun(A,bool)] :
      ( ( X = Ya )
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),image2(D,A,Ya),top_top(set(D)))))
           => ( pp(aa(A,bool,P2,Z3))
            <=> pp(aa(A,bool,Pa,Z3)) ) )
       => ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),P2),X))
        <=> pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),Pa),Ya)) ) ) ) ).

% fun.pred_cong
tff(fact_7137_fun_Opred__mono__strong,axiom,
    ! [A: $tType,D: $tType,P2: fun(A,bool),X: fun(D,A),Pa: fun(A,bool)] :
      ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),P2),X))
     => ( ! [Z3: A] :
            ( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),image2(D,A,X),top_top(set(D)))))
           => ( pp(aa(A,bool,P2,Z3))
             => pp(aa(A,bool,Pa,Z3)) ) )
       => pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),Pa),X)) ) ) ).

% fun.pred_mono_strong
tff(fact_7138_fun_Opred__rel,axiom,
    ! [A: $tType,D: $tType,P2: fun(A,bool),X: fun(D,A)] :
      ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),P2),X))
    <=> pp(aa(fun(D,A),bool,aa(fun(D,A),fun(fun(D,A),bool),bNF_rel_fun(D,D,A,A,fequal(D),bNF_eq_onp(A,P2)),X),X)) ) ).

% fun.pred_rel
tff(fact_7139_fun_ODomainp__rel,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : aa(fun(fun(C,A),fun(fun(C,B),bool)),fun(fun(C,A),bool),domainp(fun(C,A),fun(C,B)),bNF_rel_fun(C,C,A,B,fequal(C),R2)) = aa(fun(A,bool),fun(fun(C,A),bool),basic_pred_fun(C,A,aTP_Lamp_apf(C,bool)),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R2)) ).

% fun.Domainp_rel
tff(fact_7140_fun_Opred__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,Q2: fun(B,bool),F3: fun(A,B),X: fun(D,A)] :
      ( pp(aa(fun(D,B),bool,aa(fun(B,bool),fun(fun(D,B),bool),basic_pred_fun(D,B,aTP_Lamp_apd(D,bool)),Q2),aa(fun(D,A),fun(D,B),comp(A,B,D,F3),X)))
    <=> pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),aa(fun(A,B),fun(A,bool),comp(B,bool,A,Q2),F3)),X)) ) ).

% fun.pred_map
tff(fact_7141_fun_Opred__set,axiom,
    ! [D: $tType,A: $tType,P2: fun(A,bool),X5: fun(D,A)] :
      ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),P2),X5))
    <=> ! [Xa3: A] :
          ( pp(aa(set(A),bool,member(A,Xa3),aa(set(D),set(A),image2(D,A,X5),top_top(set(D)))))
         => pp(aa(A,bool,P2,Xa3)) ) ) ).

% fun.pred_set
tff(fact_7142_fun_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(fun(D,B),bool)),bool,aa(fun(fun(A,bool),fun(fun(D,A),bool)),fun(fun(fun(B,bool),fun(fun(D,B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(fun(D,A),bool),fun(fun(D,B),bool),bNF_rel_fun(A,B,bool,bool,R2,fequal(bool)),bNF_rel_fun(fun(D,A),fun(D,B),bool,bool,bNF_rel_fun(D,D,A,B,fequal(D),R2),fequal(bool))),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool))),basic_pred_fun(D,B,aTP_Lamp_apd(D,bool)))) ).

% fun.pred_transfer
tff(fact_7143_fun_Orel__eq__onp,axiom,
    ! [D: $tType,A: $tType,P2: fun(A,bool)] : bNF_rel_fun(D,D,A,A,fequal(D),bNF_eq_onp(A,P2)) = bNF_eq_onp(fun(D,A),aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_apd(D,bool)),P2)) ).

% fun.rel_eq_onp
tff(fact_7144_Inter__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A6)
     => ( bi_total(A,B,A6)
       => pp(aa(fun(set(set(B)),set(B)),bool,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),bool),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A6)),bNF_rel_set(A,B,A6)),complete_Inf_Inf(set(A))),complete_Inf_Inf(set(B)))) ) ) ).

% Inter_transfer
tff(fact_7145_Rep__unit,axiom,
    ! [X: product_unit] : pp(aa(set(bool),bool,member(bool,aa(product_unit,bool,product_Rep_unit,X)),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool))))) ).

% Rep_unit
tff(fact_7146_strict__subset__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A6)
     => pp(aa(fun(set(B),fun(set(B),bool)),bool,aa(fun(set(A),fun(set(A),bool)),fun(fun(set(B),fun(set(B),bool)),bool),bNF_rel_fun(set(A),set(B),fun(set(A),bool),fun(set(B),bool),bNF_rel_set(A,B,A6),bNF_rel_fun(set(A),set(B),bool,bool,bNF_rel_set(A,B,A6),fequal(bool))),ord_less(set(A))),ord_less(set(B)))) ) ).

% strict_subset_transfer
tff(fact_7147_subset__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A6)
     => pp(aa(fun(set(B),fun(set(B),bool)),bool,aa(fun(set(A),fun(set(A),bool)),fun(fun(set(B),fun(set(B),bool)),bool),bNF_rel_fun(set(A),set(B),fun(set(A),bool),fun(set(B),bool),bNF_rel_set(A,B,A6),bNF_rel_fun(set(A),set(B),bool,bool,bNF_rel_set(A,B,A6),fequal(bool))),ord_less_eq(set(A))),ord_less_eq(set(B)))) ) ).

% subset_transfer
tff(fact_7148_typedef__bi__unique,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A6: set(A),T8: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A6)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => bi_unique(A,B,T8) ) ) ).

% typedef_bi_unique
tff(fact_7149_Rep__unit__inject,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( pp(aa(product_unit,bool,product_Rep_unit,X))
      <=> pp(aa(product_unit,bool,product_Rep_unit,Y)) )
    <=> ( X = Y ) ) ).

% Rep_unit_inject
tff(fact_7150_Rep__unit__induct,axiom,
    ! [Y: bool,P2: fun(bool,bool)] :
      ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ( ! [X3: product_unit] : pp(aa(bool,bool,P2,aa(product_unit,bool,product_Rep_unit,X3)))
       => pp(aa(bool,bool,P2,Y)) ) ) ).

% Rep_unit_induct
tff(fact_7151_Rep__unit__cases,axiom,
    ! [Y: bool] :
      ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ~ ! [X3: product_unit] :
            ( pp(Y)
          <=> ~ pp(aa(product_unit,bool,product_Rep_unit,X3)) ) ) ).

% Rep_unit_cases
tff(fact_7152_right__total__Inter__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A6)
     => ( right_total(A,B,A6)
       => pp(aa(fun(set(set(B)),set(B)),bool,aa(fun(set(set(A)),set(A)),fun(fun(set(set(B)),set(B)),bool),bNF_rel_fun(set(set(A)),set(set(B)),set(A),set(B),bNF_rel_set(set(A),set(B),bNF_rel_set(A,B,A6)),bNF_rel_set(A,B,A6)),aTP_Lamp_apg(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),A6)),complete_Inf_Inf(set(B)))) ) ) ).

% right_total_Inter_transfer
tff(fact_7153_type__definition__unit,axiom,
    type_definition(product_unit,bool,product_Rep_unit,product_Abs_unit,aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))) ).

% type_definition_unit
tff(fact_7154_Unity__def,axiom,
    product_Unity = aa(bool,product_unit,product_Abs_unit,fTrue) ).

% Unity_def
tff(fact_7155_typedef__right__total,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A6: set(A),T8: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A6)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => right_total(A,B,T8) ) ) ).

% typedef_right_total
tff(fact_7156_Rep__unit__inverse,axiom,
    ! [X: product_unit] : aa(bool,product_unit,product_Abs_unit,aa(product_unit,bool,product_Rep_unit,X)) = X ).

% Rep_unit_inverse
tff(fact_7157_right__total__Collect__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( right_total(A,B,A6)
     => pp(aa(fun(fun(B,bool),set(B)),bool,aa(fun(fun(A,bool),set(A)),fun(fun(fun(B,bool),set(B)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),set(A),set(B),bNF_rel_fun(A,B,bool,bool,A6,fequal(bool)),bNF_rel_set(A,B,A6)),aTP_Lamp_api(fun(A,fun(B,bool)),fun(fun(A,bool),set(A)),A6)),collect(B))) ) ).

% right_total_Collect_transfer
tff(fact_7158_right__total__Domainp__transfer,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,B5: fun(A,fun(B,bool)),A6: fun(C,fun(D,bool))] :
      ( right_total(A,B,B5)
     => pp(aa(fun(fun(D,fun(B,bool)),fun(D,bool)),bool,aa(fun(fun(C,fun(A,bool)),fun(C,bool)),fun(fun(fun(D,fun(B,bool)),fun(D,bool)),bool),bNF_rel_fun(fun(C,fun(A,bool)),fun(D,fun(B,bool)),fun(C,bool),fun(D,bool),bNF_rel_fun(C,D,fun(A,bool),fun(B,bool),A6,bNF_rel_fun(A,B,bool,bool,B5,fequal(bool))),bNF_rel_fun(C,D,bool,bool,A6,fequal(bool))),aTP_Lamp_apj(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(C,bool)),B5)),domainp(D,B))) ) ).

% right_total_Domainp_transfer
tff(fact_7159_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: bool] :
          ( ( X = aa(bool,product_unit,product_Abs_unit,Y3) )
         => ~ pp(aa(set(bool),bool,member(bool,Y3),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool))))) ) ).

% Abs_unit_cases
tff(fact_7160_Abs__unit__induct,axiom,
    ! [P2: fun(product_unit,bool),X: product_unit] :
      ( ! [Y3: bool] :
          ( pp(aa(set(bool),bool,member(bool,Y3),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
         => pp(aa(product_unit,bool,P2,aa(bool,product_unit,product_Abs_unit,Y3))) )
     => pp(aa(product_unit,bool,P2,X)) ) ).

% Abs_unit_induct
tff(fact_7161_Abs__unit__inject,axiom,
    ! [X: bool,Y: bool] :
      ( pp(aa(set(bool),bool,member(bool,X),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
       => ( ( aa(bool,product_unit,product_Abs_unit,X) = aa(bool,product_unit,product_Abs_unit,Y) )
        <=> ( pp(X)
          <=> pp(Y) ) ) ) ) ).

% Abs_unit_inject
tff(fact_7162_right__total__Compl__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A6)
     => ( right_total(A,B,A6)
       => pp(aa(fun(set(B),set(B)),bool,aa(fun(set(A),set(A)),fun(fun(set(B),set(B)),bool),bNF_rel_fun(set(A),set(B),set(A),set(B),bNF_rel_set(A,B,A6),bNF_rel_set(A,B,A6)),aTP_Lamp_apk(fun(A,fun(B,bool)),fun(set(A),set(A)),A6)),uminus_uminus(set(B)))) ) ) ).

% right_total_Compl_transfer
tff(fact_7163_right__total__fun__eq__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] :
      ( right_total(A,B,A6)
     => ( bi_unique(C,D,B5)
       => pp(aa(fun(fun(B,D),fun(fun(B,D),bool)),bool,aa(fun(fun(A,C),fun(fun(A,C),bool)),fun(fun(fun(B,D),fun(fun(B,D),bool)),bool),bNF_rel_fun(fun(A,C),fun(B,D),fun(fun(A,C),bool),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A6,B5),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A6,B5),fequal(bool))),aTP_Lamp_apl(fun(A,fun(B,bool)),fun(fun(A,C),fun(fun(A,C),bool)),A6)),fequal(fun(B,D)))) ) ) ).

% right_total_fun_eq_transfer
tff(fact_7164_vimage__right__total__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,B5: fun(A,fun(B,bool)),A6: fun(C,fun(D,bool))] :
      ( bi_unique(A,B,B5)
     => ( right_total(C,D,A6)
       => pp(aa(fun(fun(D,B),fun(set(B),set(D))),bool,aa(fun(fun(C,A),fun(set(A),set(C))),fun(fun(fun(D,B),fun(set(B),set(D))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(set(A),set(C)),fun(set(B),set(D)),bNF_rel_fun(C,D,A,B,A6,B5),bNF_rel_fun(set(A),set(B),set(C),set(D),bNF_rel_set(A,B,B5),bNF_rel_set(C,D,A6))),aTP_Lamp_apm(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),A6)),vimage(D,B))) ) ) ).

% vimage_right_total_transfer
tff(fact_7165_Abs__unit__inverse,axiom,
    ! [Y: bool] :
      ( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),aa(bool,fun(set(bool),set(bool)),insert(bool),fTrue),bot_bot(set(bool)))))
     => ( pp(aa(product_unit,bool,product_Rep_unit,aa(bool,product_unit,product_Abs_unit,Y)))
      <=> pp(Y) ) ) ).

% Abs_unit_inverse
tff(fact_7166_in__measures_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat),Fs: list(fun(A,nat))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
        | ( ( aa(A,nat,F3,X) = aa(A,nat,F3,Y) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs))) ) ) ) ).

% in_measures(2)
tff(fact_7167_select__def,axiom,
    ! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(nat,code_natural,code_natural_of_nat,aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_apn(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).

% select_def
tff(fact_7168_in__measures_I1_J,axiom,
    ! [A: $tType,X: A,Y: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,nil(fun(A,nat))))) ).

% in_measures(1)
tff(fact_7169_plus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% plus_natural.rep_eq
tff(fact_7170_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),X),Xa2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2))) ) ).

% less_natural.rep_eq
tff(fact_7171_less__eq__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),Xa2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2))) ) ).

% less_eq_natural.rep_eq
tff(fact_7172_plus__natural__def,axiom,
    plus_plus(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),plus_plus(nat)) ).

% plus_natural_def
tff(fact_7173_less__natural__def,axiom,
    ord_less(code_natural) = aa(fun(nat,fun(nat,bool)),fun(code_natural,fun(code_natural,bool)),map_fun(code_natural,nat,fun(nat,bool),fun(code_natural,bool),code_nat_of_natural,map_fun(code_natural,nat,bool,bool,code_nat_of_natural,id(bool))),ord_less(nat)) ).

% less_natural_def
tff(fact_7174_less__eq__natural__def,axiom,
    ord_less_eq(code_natural) = aa(fun(nat,fun(nat,bool)),fun(code_natural,fun(code_natural,bool)),map_fun(code_natural,nat,fun(nat,bool),fun(code_natural,bool),code_nat_of_natural,map_fun(code_natural,nat,bool,bool,code_nat_of_natural,id(bool))),ord_less_eq(nat)) ).

% less_eq_natural_def
tff(fact_7175_measures__less,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs)))) ) ).

% measures_less
tff(fact_7176_measures__lesseq,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs)))) ) ) ).

% measures_lesseq
tff(fact_7177_measures__def,axiom,
    ! [A: $tType,Fs: list(fun(A,nat))] : measures(A,Fs) = inv_image(list(nat),A,lex(nat,less_than),aTP_Lamp_apo(list(fun(A,nat)),fun(A,list(nat)),Fs)) ).

% measures_def
tff(fact_7178_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_zero(code_natural) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,N)),aa(nat,nat,suc,zero_zero(nat)))),aa(code_natural,nat,code_nat_of_natural,N))) ) ).

% natural_decr
tff(fact_7179_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [N: int,X: A] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( power_int(A,X,N) = aa(nat,A,power_power(A,X),aa(int,nat,nat2,N)) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( power_int(A,X,N) = aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N))) ) ) ) ) ).

% power_int_def
tff(fact_7180_repeat__mset__distrib__add__mset,axiom,
    ! [A: $tType,N: nat,A3: A,A6: multiset(A)] : aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),N),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A6)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),replicate_mset(A,N,A3)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),N),A6)) ).

% repeat_mset_distrib_add_mset
tff(fact_7181_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).

% power_int_add_numeral2
tff(fact_7182_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),power_int(A,X,aa(num,int,numeral_numeral(int),N))) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% power_int_add_numeral
tff(fact_7183_in__replicate__mset,axiom,
    ! [A: $tType,X: A,N: nat,Y: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(multiset(A),set(A),set_mset(A),replicate_mset(A,N,Y))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
        & ( X = Y ) ) ) ).

% in_replicate_mset
tff(fact_7184_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,B2,N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_7185_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N7))) ) ) ) ).

% power_int_increasing
tff(fact_7186_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_le_power_int
tff(fact_7187_msubseteq__replicate__msetE,axiom,
    ! [A: $tType,A6: multiset(A),N: nat,A3: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A6),replicate_mset(A,N,A3)))
     => ~ ! [M5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N))
           => ( A6 != replicate_mset(A,M5,A3) ) ) ) ).

% msubseteq_replicate_msetE
tff(fact_7188_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,A3,N7))) ) ) ) ).

% power_int_strict_increasing
tff(fact_7189_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_less_power_int
tff(fact_7190_image__mset__const__eq,axiom,
    ! [B: $tType,A: $tType,C3: A,M6: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_or(A,fun(B,A)),C3)),M6) = replicate_mset(A,aa(multiset(B),nat,size_size(multiset(B)),M6),C3) ).

% image_mset_const_eq
tff(fact_7191_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N7)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_7192_filter__eq__replicate__mset,axiom,
    ! [A: $tType,X: A,D5: multiset(A)] : aa(multiset(A),multiset(A),aa(fun(A,bool),fun(multiset(A),multiset(A)),filter_mset(A),aTP_Lamp_ap(A,fun(A,bool),X)),D5) = replicate_mset(A,aa(A,nat,aa(multiset(A),fun(A,nat),count(A),D5),X),X) ).

% filter_eq_replicate_mset
tff(fact_7193_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),power_int(A,Y,N))) ) ) ) ) ).

% power_int_mono
tff(fact_7194_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_7195_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X,N))) ) ) ) ).

% one_le_power_int
tff(fact_7196_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),power_int(A,A3,N))) ) ) ) ).

% one_less_power_int
tff(fact_7197_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).

% power_int_add
tff(fact_7198_replicate__mset__msubseteq__iff,axiom,
    ! [A: $tType,M: nat,A3: A,N: nat,B2: A] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),replicate_mset(A,M,A3)),replicate_mset(A,N,B2)))
    <=> ( ( M = zero_zero(nat) )
        | ( ( A3 = B2 )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% replicate_mset_msubseteq_iff
tff(fact_7199_count__le__replicate__mset__subset__eq,axiom,
    ! [A: $tType,N: nat,M6: multiset(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M6),X)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),replicate_mset(A,N,X)),M6)) ) ).

% count_le_replicate_mset_subset_eq
tff(fact_7200_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,B2,N))) ) ) ) ) ).

% power_int_strict_mono
tff(fact_7201_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_antimono
tff(fact_7202_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N7: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N7))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => ( ( ( A3 != zero_zero(A) )
                  | ( N7 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N7)),power_int(A,A3,N))) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_7203_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),one_one(A))) ) ) ) ) ).

% power_int_le_one
tff(fact_7204_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N)) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_7205_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X,M)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N)) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_7206_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M)) ) ) ) ).

% power_int_add_1'
tff(fact_7207_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),X) ) ) ) ).

% power_int_add_1
tff(fact_7208_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),field2(A,R3)))
       => ( ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),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))),minus_minus(set(product_prod(A,A))),R3),id2(A))))
              | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),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))),minus_minus(set(product_prod(A,A))),R3),id2(A)))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( ( A3 = B2 )
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_7209_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B5: set(A),F4: fun(A,filter(B)),P2: fun(B,bool)] :
      ( ( B5 != bot_bot(set(A)) )
     => ( ! [A5: A] :
            ( pp(aa(set(A),bool,member(A,A5),B5))
           => ! [B4: A] :
                ( pp(aa(set(A),bool,member(A,B4),B5))
               => ? [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),B5))
                    & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F4,X5)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A5)),aa(A,filter(B),F4,B4)))) ) ) )
       => ( eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B5)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),B5))
              & eventually(B,P2,aa(A,filter(B),F4,X4)) ) ) ) ) ).

% eventually_INF_base
tff(fact_7210_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P2: bool] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_zu(bool,fun(A,bool),P2),F4)
      <=> pp(P2) ) ) ).

% eventually_const
tff(fact_7211_eventually__sequentially__Suc,axiom,
    ! [P2: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_ajv(fun(nat,bool),fun(nat,bool),P2),at_top(nat))
    <=> eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_7212_eventually__sequentially__seg,axiom,
    ! [P2: fun(nat,bool),K2: nat] :
      ( eventually(nat,aa(nat,fun(nat,bool),aTP_Lamp_app(fun(nat,bool),fun(nat,fun(nat,bool)),P2),K2),at_top(nat))
    <=> eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_7213_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A6: set(A),P2: fun(set(A),bool)] :
      ( ! [X10: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X10))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X10),A6))
           => pp(aa(set(A),bool,P2,X10)) ) )
     => eventually(set(A),P2,finite5375528669736107172at_top(A,A6)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_7214_eventually__inf__principal,axiom,
    ! [A: $tType,P2: fun(A,bool),F4: filter(A),S2: set(A)] :
      ( eventually(A,P2,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),principal(A,S2)))
    <=> eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_apq(fun(A,bool),fun(set(A),fun(A,bool)),P2),S2),F4) ) ).

% eventually_inf_principal
tff(fact_7215_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_adq(fun(A,fun(B,bool)),fun(A,bool),P2),F4)
    <=> ? [Y9: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_apr(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),P2),Y9),F4) ) ).

% eventually_ex
tff(fact_7216_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
         => ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = B2 ) )
        & ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
         => ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = A3 ) ) ) ) ).

% wo_rel.max2_def
tff(fact_7217_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ! [X3: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != X3 )
                  & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3)),R3)) )
               => pp(aa(A,bool,P2,Y5)) )
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,A3)) ) ) ).

% wo_rel.well_order_induct
tff(fact_7218_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ! [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),field2(A,R3)))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,member(A,Xa),field2(A,R3)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),R3))
                | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X5)),R3)) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_7219_well__order__induct__imp,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ! [X3: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != X3 )
                  & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3)),R3)) )
               => ( pp(aa(set(A),bool,member(A,Y5),field2(A,R3)))
                 => pp(aa(A,bool,P2,Y5)) ) )
           => ( pp(aa(set(A),bool,member(A,X3),field2(A,R3)))
             => pp(aa(A,bool,P2,X3)) ) )
       => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
         => pp(aa(A,bool,P2,A3)) ) ) ) ).

% well_order_induct_imp
tff(fact_7220_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R3)))
         => ( ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = A3 )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3)) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_7221_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R3)))
         => ( ( bNF_We1388413361240627857o_max2(A,R3,A3,B2) = B2 )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_7222_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3)) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_7223_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),S: set(B),F4: filter(A)] :
      ( filterlim(A,B,F3,principal(B,S),F4)
    <=> eventually(A,aa(set(B),fun(A,bool),aTP_Lamp_afg(fun(A,B),fun(set(B),fun(A,bool)),F3),S),F4) ) ).

% filterlim_principal
tff(fact_7224_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,I2: A,I5: set(A),P2: fun(B,bool),F4: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,member(A,I2),I5))
     => ( eventually(B,P2,aa(A,filter(B),F4,I2))
       => eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I5))) ) ) ).

% eventually_INF1
tff(fact_7225_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C3: A] : eventually(A,aTP_Lamp_aps(A,fun(A,bool),C3),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_7226_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_fq(nat,bool),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_7227_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_top(A))
        <=> ? [N13: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N13),N3))
             => pp(aa(A,bool,P2,N3)) ) ) ) ).

% eventually_at_top_dense
tff(fact_7228_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C3: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),C3),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_7229_eventually__all__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(B)
     => ! [P2: fun(A,fun(B,bool)),Net: filter(A)] :
          ( ! [Y3: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_apt(fun(A,fun(B,bool)),fun(B,fun(A,bool)),P2),Y3),Net)
         => eventually(A,aTP_Lamp_apu(fun(A,fun(B,bool)),fun(A,bool),P2),Net) ) ) ).

% eventually_all_finite
tff(fact_7230_eventually__sup,axiom,
    ! [A: $tType,P2: fun(A,bool),F4: filter(A),F6: filter(A)] :
      ( eventually(A,P2,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F6))
    <=> ( eventually(A,P2,F4)
        & eventually(A,P2,F6) ) ) ).

% eventually_sup
tff(fact_7231_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C3: A] : eventually(A,aTP_Lamp_apv(A,fun(A,bool),C3),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_7232_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_bot(A))
        <=> ? [N13: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N3),N13))
             => pp(aa(A,bool,P2,N3)) ) ) ) ).

% eventually_at_bot_dense
tff(fact_7233_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C3: A] : eventually(A,aTP_Lamp_apw(A,fun(A,bool),C3),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_7234_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C4: bool,P2: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_apx(bool,fun(fun(A,bool),fun(A,bool)),C4),P2),F4)
    <=> ( pp(C4)
       => eventually(A,P2,F4) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_7235_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C4: bool,P2: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_apy(bool,fun(fun(A,bool),fun(A,bool)),C4),P2),F4)
    <=> ( pp(C4)
        | eventually(A,P2,F4) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_7236_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P2: fun(A,bool),C4: bool,F4: filter(A)] :
      ( eventually(A,aa(bool,fun(A,bool),aTP_Lamp_apz(fun(A,bool),fun(bool,fun(A,bool)),P2),C4),F4)
    <=> ( eventually(A,P2,F4)
        | pp(C4) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_7237_eventually__mp,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_dd(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2),F4)
     => ( eventually(A,P2,F4)
       => eventually(A,Q2,F4) ) ) ).

% eventually_mp
tff(fact_7238_eventually__True,axiom,
    ! [A: $tType,F4: filter(A)] : eventually(A,aTP_Lamp_av(A,bool),F4) ).

% eventually_True
tff(fact_7239_eventually__conj,axiom,
    ! [A: $tType,P2: fun(A,bool),F4: filter(A),Q2: fun(A,bool)] :
      ( eventually(A,P2,F4)
     => ( eventually(A,Q2,F4)
       => eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ah(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2),F4) ) ) ).

% eventually_conj
tff(fact_7240_eventually__elim2,axiom,
    ! [A: $tType,P2: fun(A,bool),F4: filter(A),Q2: fun(A,bool),R2: fun(A,bool)] :
      ( eventually(A,P2,F4)
     => ( eventually(A,Q2,F4)
       => ( ! [I3: A] :
              ( pp(aa(A,bool,P2,I3))
             => ( pp(aa(A,bool,Q2,I3))
               => pp(aa(A,bool,R2,I3)) ) )
         => eventually(A,R2,F4) ) ) ) ).

% eventually_elim2
tff(fact_7241_eventually__subst,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aqa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2),F4)
     => ( eventually(A,P2,F4)
      <=> eventually(A,Q2,F4) ) ) ).

% eventually_subst
tff(fact_7242_eventually__rev__mp,axiom,
    ! [A: $tType,P2: fun(A,bool),F4: filter(A),Q2: fun(A,bool)] :
      ( eventually(A,P2,F4)
     => ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_dd(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2),F4)
       => eventually(A,Q2,F4) ) ) ).

% eventually_rev_mp
tff(fact_7243_eventually__conj__iff,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool),F4: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ah(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2),F4)
    <=> ( eventually(A,P2,F4)
        & eventually(A,Q2,F4) ) ) ).

% eventually_conj_iff
tff(fact_7244_not__eventually__impI,axiom,
    ! [A: $tType,P2: fun(A,bool),F4: filter(A),Q2: fun(A,bool)] :
      ( eventually(A,P2,F4)
     => ( ~ eventually(A,Q2,F4)
       => ~ eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_dd(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2),F4) ) ) ).

% not_eventually_impI
tff(fact_7245_False__imp__not__eventually,axiom,
    ! [A: $tType,P2: fun(A,bool),Net: filter(A)] :
      ( ! [X3: A] : ~ pp(aa(A,bool,P2,X3))
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P2,Net) ) ) ).

% False_imp_not_eventually
tff(fact_7246_eventually__const__iff,axiom,
    ! [A: $tType,P2: bool,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_zu(bool,fun(A,bool),P2),F4)
    <=> ( pp(P2)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_7247_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_aq(A,bool),F4) ) ).

% trivial_limit_def
tff(fact_7248_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A6: set(A),P2: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aqb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A6),P2),Net)
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P2),X4),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_7249_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),P2: fun(B,fun(A,bool)),Net: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A6))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P2),X3),Net) )
       => eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aqb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A6),P2),Net) ) ) ).

% eventually_ball_finite
tff(fact_7250_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),F4: filter(A),F3: fun(B,A),G5: filter(B)] :
      ( eventually(A,P2,F4)
     => ( filterlim(B,A,F3,F4,G5)
       => eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqc(fun(A,bool),fun(fun(B,A),fun(B,bool)),P2),F3),G5) ) ) ).

% eventually_compose_filterlim
tff(fact_7251_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F14: filter(A),F15: filter(A),F23: filter(B),F24: filter(B),F3: fun(B,A),G3: fun(B,A)] :
      ( ( F14 = F15 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqd(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G3),F23)
         => ( filterlim(B,A,F3,F14,F23)
          <=> filterlim(B,A,G3,F15,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_7252_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F23: filter(B),F14: filter(A)] :
      ( filterlim(A,B,F3,F23,F14)
    <=> ! [P8: fun(B,bool)] :
          ( eventually(B,P8,F23)
         => eventually(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_afe(fun(A,B),fun(fun(B,bool),fun(A,bool)),F3),P8),F14) ) ) ).

% filterlim_iff
tff(fact_7253_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(B),G5: filter(A),F6: filter(B),G8: filter(A),F5: fun(A,B)] :
      ( filterlim(A,B,F3,F4,G5)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F4),F6))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G8),G5))
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_ape(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),F5),G8)
           => filterlim(A,B,F5,F6,G8) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_7254_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,bool),aTP_Lamp_mo(A,fun(A,bool)),C3),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_7255_filter__leD,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A),P2: fun(A,bool)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F6))
     => ( eventually(A,P2,F6)
       => eventually(A,P2,F4) ) ) ).

% filter_leD
tff(fact_7256_filter__leI,axiom,
    ! [A: $tType,F6: filter(A),F4: filter(A)] :
      ( ! [P: fun(A,bool)] :
          ( eventually(A,P,F6)
         => eventually(A,P,F4) )
     => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F6)) ) ).

% filter_leI
tff(fact_7257_le__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F6))
    <=> ! [P8: fun(A,bool)] :
          ( eventually(A,P8,F6)
         => eventually(A,P8,F4) ) ) ).

% le_filter_def
tff(fact_7258_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_bot(A))
        <=> ? [N13: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N3),N13))
             => pp(aa(A,bool,P2,N3)) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_7259_eventually__sequentially,axiom,
    ! [P2: fun(nat,bool)] :
      ( eventually(nat,P2,at_top(nat))
    <=> ? [N13: nat] :
        ! [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N13),N3))
         => pp(aa(nat,bool,P2,N3)) ) ) ).

% eventually_sequentially
tff(fact_7260_eventually__sequentiallyI,axiom,
    ! [C3: nat,P2: fun(nat,bool)] :
      ( ! [X3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C3),X3))
         => pp(aa(nat,bool,P2,X3)) )
     => eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_7261_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_top(A))
        <=> ? [N13: A] :
            ! [N3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N13),N3))
             => pp(aa(A,bool,P2,N3)) ) ) ) ).

% eventually_at_top_linorder
tff(fact_7262_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,P2: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),X3))
             => pp(aa(A,bool,P2,X3)) )
         => eventually(A,P2,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_7263_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( pp(aa(filter(nat),bool,aa(filter(nat),fun(filter(nat),bool),ord_less_eq(filter(nat)),F4),at_top(nat)))
    <=> ! [N13: nat] : eventually(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N13),F4) ) ).

% le_sequentially
tff(fact_7264_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,bool),ord_less_eq(A),C3),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_7265_le__principal,axiom,
    ! [A: $tType,F4: filter(A),A6: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),principal(A,A6)))
    <=> eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6),F4) ) ).

% le_principal
tff(fact_7266_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q2: fun(A,bool),F3: fun(A,B),P2: fun(B,bool),G3: fun(B,A)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q2,X3))
             => ( pp(aa(A,bool,Q2,Y3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P2,X3))
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P2,X3))
                 => pp(aa(A,bool,Q2,aa(B,A,G3,X3))) )
             => ( eventually(A,Q2,at_top(A))
               => ( eventually(B,P2,at_top(B))
                 => filterlim(A,B,F3,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_7267_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P2: fun(set(A),bool),A6: set(A)] :
      ( eventually(set(A),P2,finite5375528669736107172at_top(A,A6))
    <=> ? [X14: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),X14))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X14),A6))
          & ! [Y9: set(A)] :
              ( ( pp(aa(set(A),bool,finite_finite2(A),Y9))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X14),Y9))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y9),A6)) )
             => pp(aa(set(A),bool,P2,Y9)) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_7268_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_top(A))
         => eventually(A,aTP_Lamp_aqe(fun(A,bool),fun(A,bool),P2),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_7269_eventually__Inf__base,axiom,
    ! [A: $tType,B5: set(filter(A)),P2: fun(A,bool)] :
      ( ( B5 != bot_bot(set(filter(A))) )
     => ( ! [F7: filter(A)] :
            ( pp(aa(set(filter(A)),bool,member(filter(A),F7),B5))
           => ! [G6: filter(A)] :
                ( pp(aa(set(filter(A)),bool,member(filter(A),G6),B5))
               => ? [X5: filter(A)] :
                    ( pp(aa(set(filter(A)),bool,member(filter(A),X5),B5))
                    & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),X5),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F7),G6))) ) ) )
       => ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
        <=> ? [X4: filter(A)] :
              ( pp(aa(set(filter(A)),bool,member(filter(A),X4),B5))
              & eventually(A,P2,X4) ) ) ) ) ).

% eventually_Inf_base
tff(fact_7270_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),F4: filter(B),G3: fun(B,A)] :
          ( filterlim(B,A,F3,at_top(A),F4)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqf(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G3),F4)
           => filterlim(B,A,G3,at_top(A),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_7271_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C3),Z9))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_aqg(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_7272_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_aqg(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_top
tff(fact_7273_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_aqh(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_7274_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),P2: fun(B,bool),F4: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),A6)))
      <=> ? [Q9: fun(A,fun(B,bool))] :
            ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),A6))
               => eventually(B,aa(A,fun(B,bool),Q9,X4),aa(A,filter(B),F4,X4)) )
            & ! [Y4: B] :
                ( ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A6))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q9,X4),Y4)) )
               => pp(aa(B,bool,P2,Y4)) ) ) ) ) ).

% eventually_INF_finite
tff(fact_7275_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Z9),C3))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_aqi(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_7276_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_aqi(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_bot
tff(fact_7277_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_aqj(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_7278_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),field2(A,R3)))
       => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3))
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_7279_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_wo_rel
tff(fact_7280_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R3)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R3,A3,B2))),R3))
            & pp(aa(set(A),bool,member(A,bNF_We1388413361240627857o_max2(A,R3,A3,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_7281_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C3),Z9))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_aqk(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7282_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),F4: fun(B,filter(A)),B5: set(B)] :
      ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),B5)))
    <=> ? [X14: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X14),B5))
          & pp(aa(set(B),bool,finite_finite2(B),X14))
          & eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),X14))) ) ) ).

% eventually_INF
tff(fact_7283_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z9: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z9),C3))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_aql(fun(A,B),fun(B,fun(A,bool)),F3),Z9),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_7284_eventually__Inf,axiom,
    ! [A: $tType,P2: fun(A,bool),B5: set(filter(A))] :
      ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
    <=> ? [X14: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X14),B5))
          & pp(aa(set(filter(A)),bool,finite_finite2(filter(A)),X14))
          & eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X14)) ) ) ).

% eventually_Inf
tff(fact_7285_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,set(B)),A6: set(B),F4: filter(A)] :
      ( filterlim(A,set(B),F3,finite5375528669736107172at_top(B,A6),F4)
    <=> ! [X14: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),X14))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X14),A6)) )
         => eventually(A,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_aqm(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),F3),A6),X14),F4) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_7286_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G3: fun(B,A),Y6: set(B),X6: set(A),F4: filter(B),F3: fun(A,C)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G3),Y6)),X6))
     => ( eventually(B,aTP_Lamp_be(set(B),fun(B,bool),Y6),F4)
       => ( map_filter_on(A,C,X6,F3,map_filter_on(B,A,Y6,G3,F4)) = map_filter_on(B,C,Y6,aa(fun(B,A),fun(B,C),comp(A,C,B,F3),G3),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_7287_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( ( B5 != bot_bot(set(A)) )
         => ? [X_1: A] : pp(aa(A,bool,bNF_We4791949203932849705sMinim(A,R3,B5),X_1)) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_7288_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F4: filter(A),P2: fun(B,bool),F3: fun(A,B)] :
      ( eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6),F4)
     => ( eventually(B,P2,map_filter_on(A,B,X6,F3,F4))
      <=> eventually(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),aTP_Lamp_aqn(set(A),fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),X6),P2),F3),F4) ) ) ).

% eventually_map_filter_on
tff(fact_7289_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(A,bool,bNF_We4791949203932849705sMinim(A,R3,A6),B2))
      <=> ( pp(aa(set(A),bool,member(A,B2),A6))
          & ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A6))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X4)),R3)) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_7290_wo__rel_Ominim__isMinim,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( ( B5 != bot_bot(set(A)) )
         => pp(aa(A,bool,bNF_We4791949203932849705sMinim(A,R3,B5),bNF_We6954850376910717587_minim(A,R3,B5))) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_7291_wo__rel_Ominim__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( bNF_We6954850376910717587_minim(A,R3,A6) = the(A,bNF_We4791949203932849705sMinim(A,R3,A6)) ) ) ).

% wo_rel.minim_def
tff(fact_7292_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,A3),B5))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,member(A,B4),B5))
               => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B4)),R3)) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R3,B5) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_7293_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),B5))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R3,B5)),B2)),R3)) ) ) ) ).

% wo_rel.minim_least
tff(fact_7294_wo__rel_Ominim__in,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( ( B5 != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_We6954850376910717587_minim(A,R3,B5)),B5)) ) ) ) ).

% wo_rel.minim_in
tff(fact_7295_wo__rel_Ominim__inField,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( ( B5 != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_We6954850376910717587_minim(A,R3,B5)),field2(A,R3))) ) ) ) ).

% wo_rel.minim_inField
tff(fact_7296_Sup__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S) = abs_filter(A,aTP_Lamp_aqo(set(filter(A)),fun(fun(A,bool),bool),S)) ).

% Sup_filter_def
tff(fact_7297_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),linord329482645794927042rt_key(B,A,F3,X,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_7298_insort__insert__triv,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_pk(A,A),X,Xs) = Xs ) ) ) ).

% insort_insert_triv
tff(fact_7299_top__filter__def,axiom,
    ! [A: $tType] : top_top(filter(A)) = abs_filter(A,fAll(A)) ).

% top_filter_def
tff(fact_7300_bot__filter__def,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = abs_filter(A,aTP_Lamp_aqp(fun(A,bool),bool)) ).

% bot_filter_def
tff(fact_7301_sup__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F4),F6) = abs_filter(A,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_aqq(filter(A),fun(filter(A),fun(fun(A,bool),bool)),F4),F6)) ).

% sup_filter_def
tff(fact_7302_principal__def,axiom,
    ! [A: $tType,S: set(A)] : principal(A,S) = abs_filter(A,aa(set(A),fun(fun(A,bool),bool),ball(A),S)) ).

% principal_def
tff(fact_7303_map__filter__on__def,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F3: fun(A,B),F4: filter(A)] : map_filter_on(A,B,X6,F3,F4) = abs_filter(B,aa(filter(A),fun(fun(B,bool),bool),aa(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),aTP_Lamp_aqs(set(A),fun(fun(A,B),fun(filter(A),fun(fun(B,bool),bool))),X6),F3),F4)) ).

% map_filter_on_def
tff(fact_7304_set__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_pk(A,A),X,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_insert
tff(fact_7305_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),X: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_pk(A,A),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_7306_insort__insert__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ( linord329482645794927042rt_key(A,A,aTP_Lamp_pk(A,A),X,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A)),X),Xs) ) ) ) ).

% insort_insert_insort
tff(fact_7307_inf__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),F6) = abs_filter(A,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_aqt(filter(A),fun(filter(A),fun(fun(A,bool),bool)),F4),F6)) ).

% inf_filter_def
tff(fact_7308_execute__ref,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [V2: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(ref(A),ref_ref(A,V2)),H) = aa(product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aqu(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,V2,H))) ) ).

% execute_ref
tff(fact_7309_ref__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [V2: A] : ref_ref(A,V2) = heap_Time_heap(ref(A),aTP_Lamp_aqv(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),V2)) ) ).

% ref_def
tff(fact_7310_filtercomap__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F4: filter(B)] : filtercomap(A,B,F3,F4) = abs_filter(A,aa(filter(B),fun(fun(A,bool),bool),aTP_Lamp_aqw(fun(A,B),fun(filter(B),fun(fun(A,bool),bool)),F3),F4)) ).

% filtercomap_def
tff(fact_7311_admissible__chfin,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P2: fun(A,bool)] :
          ( ! [S6: set(A)] :
              ( comple1602240252501008431_chain(A,ord_less_eq(A),S6)
             => pp(aa(set(A),bool,finite_finite2(A),S6)) )
         => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P2) ) ) ).

% admissible_chfin
tff(fact_7312_eventually__filtercomapI,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),F4: filter(A),F3: fun(B,A)] :
      ( eventually(A,P2,F4)
     => eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqc(fun(A,bool),fun(fun(B,A),fun(B,bool)),P2),F3),filtercomap(B,A,F3,F4)) ) ).

% eventually_filtercomapI
tff(fact_7313_filtercomap__filtercomap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),G3: fun(B,C),F4: filter(C)] : filtercomap(A,B,F3,filtercomap(B,C,G3,F4)) = filtercomap(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqx(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),F4) ).

% filtercomap_filtercomap
tff(fact_7314_filtercomap__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filtercomap(A,A,aTP_Lamp_cc(A,A),F4) = F4 ).

% filtercomap_ident
tff(fact_7315_admissible__all,axiom,
    ! [A: $tType,B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,bool)),P2: fun(B,fun(A,bool))] :
      ( ! [Y3: A] : comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P2),Y3))
     => comple1908693960933563346ssible(B,Lub,Ord,aTP_Lamp_aqy(fun(B,fun(A,bool)),fun(B,bool),P2)) ) ).

% admissible_all
tff(fact_7316_admissible__const,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),T5: bool] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_zu(bool,fun(A,bool),T5)) ).

% admissible_const
tff(fact_7317_admissible__conj,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),P2: fun(A,bool),Q2: fun(A,bool)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P2)
     => ( comple1908693960933563346ssible(A,Lub,Ord,Q2)
       => comple1908693960933563346ssible(A,Lub,Ord,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ah(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2)) ) ) ).

% admissible_conj
tff(fact_7318_admissible__True,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool))] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_av(A,bool)) ).

% admissible_True
tff(fact_7319_admissible__ball,axiom,
    ! [A: $tType,B: $tType,A6: set(A),Lub: fun(set(B),B),Ord: fun(B,fun(B,bool)),P2: fun(B,fun(A,bool))] :
      ( ! [Y3: A] :
          ( pp(aa(set(A),bool,member(A,Y3),A6))
         => comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P2),Y3)) )
     => comple1908693960933563346ssible(B,Lub,Ord,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aqb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A6),P2)) ) ).

% admissible_ball
tff(fact_7320_filterlim__iff__le__filtercomap,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F4: filter(B),G5: filter(A)] :
      ( filterlim(A,B,F3,F4,G5)
    <=> pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G5),filtercomap(A,B,F3,F4))) ) ).

% filterlim_iff_le_filtercomap
tff(fact_7321_filtercomap__mono,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),F6: filter(A),F3: fun(B,A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F6))
     => pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtercomap(B,A,F3,F4)),filtercomap(B,A,F3,F6))) ) ).

% filtercomap_mono
tff(fact_7322_filtercomap__sup,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F14: filter(B),F23: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),filtercomap(A,B,F3,F14)),filtercomap(A,B,F3,F23))),filtercomap(A,B,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F14),F23)))) ).

% filtercomap_sup
tff(fact_7323_filtercomap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),F4: fun(C,filter(B)),B5: set(C)] : filtercomap(A,B,F3,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B5))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aqz(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F3),F4)),B5)) ).

% filtercomap_INF
tff(fact_7324_admissible__disj,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P2: fun(A,bool),Q2: fun(A,bool)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P2)
         => ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),Q2)
           => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ara(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q2)) ) ) ) ).

% admissible_disj
tff(fact_7325_eventually__filtercomap__at__top__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P2: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P2,filtercomap(B,A,F3,at_top(A)))
        <=> ? [N13: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N13),aa(B,A,F3,X4)))
             => pp(aa(B,bool,P2,X4)) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_7326_eventually__filtercomap__at__top__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P2: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P2,filtercomap(B,A,F3,at_top(A)))
        <=> ? [N13: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N13),aa(B,A,F3,X4)))
             => pp(aa(B,bool,P2,X4)) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_7327_eventually__filtercomap__at__bot__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P2: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P2,filtercomap(B,A,F3,at_bot(A)))
        <=> ? [N13: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),N13))
             => pp(aa(B,bool,P2,X4)) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_7328_eventually__filtercomap__at__bot__dense,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P2: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P2,filtercomap(B,A,F3,at_bot(A)))
        <=> ? [N13: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),N13))
             => pp(aa(B,bool,P2,X4)) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_7329_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,C),F4: fun(B,filter(C)),B5: set(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_arb(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F3),F4)),B5))),filtercomap(A,C,F3,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F4),B5))))) ).

% filtercomap_SUP
tff(fact_7330_multp__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),M6: multiset(A),N7: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),M6),N7))
    <=> pp(aa(set(product_prod(multiset(A),multiset(A))),bool,member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M6),N7)),mult(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))))) ) ).

% multp_def
tff(fact_7331_pair__lessI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T5))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T5))),fun_pair_less)) ) ) ).

% pair_lessI2
tff(fact_7332_total__pair__less,axiom,
    ! [A6: set(product_prod(nat,nat))] : total_on(product_prod(nat,nat),A6,fun_pair_less) ).

% total_pair_less
tff(fact_7333_trans__pair__less,axiom,
    trans(product_prod(nat,nat),fun_pair_less) ).

% trans_pair_less
tff(fact_7334_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z4: nat] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Z4))),fun_pair_less))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),Z4)) ) ).

% pair_less_iff1
tff(fact_7335_wf__pair__less,axiom,
    wf(product_prod(nat,nat),fun_pair_less) ).

% wf_pair_less
tff(fact_7336_pair__less__def,axiom,
    fun_pair_less = lex_prod(nat,nat,less_than,less_than) ).

% pair_less_def
tff(fact_7337_less__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M6: multiset(A),N7: multiset(A)] :
          ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),M6),N7))
        <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,ord_less(A)),M6),N7)) ) ) ).

% less_multiset_def
tff(fact_7338_pair__lessI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T5))),fun_pair_less)) ) ).

% pair_lessI1
tff(fact_7339_one__step__implies__multp,axiom,
    ! [A: $tType,J4: multiset(A),K5: multiset(A),R3: fun(A,fun(A,bool)),I5: multiset(A)] :
      ( ( J4 != zero_zero(multiset(A)) )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(multiset(A),set(A),set_mset(A),K5)))
           => ? [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),aa(multiset(A),set(A),set_mset(A),J4)))
                & pp(aa(A,bool,aa(A,fun(A,bool),R3,X3),Xa)) ) )
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),K5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I5),J4))) ) ) ).

% one_step_implies_multp
tff(fact_7340_mono__multp,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),R6: fun(A,fun(A,bool))] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R3),R6))
     => pp(aa(fun(multiset(A),fun(multiset(A),bool)),bool,aa(fun(multiset(A),fun(multiset(A),bool)),fun(fun(multiset(A),fun(multiset(A),bool)),bool),ord_less_eq(fun(multiset(A),fun(multiset(A),bool))),multp(A,R3)),multp(A,R6))) ) ).

% mono_multp
tff(fact_7341_smin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),X),XS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_strict))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert(product_prod(nat,nat)),Y),YS))),fun_min_strict)) ) ) ) ).

% smin_insertI
tff(fact_7342_pair__leqI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S2),T5))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T5))),fun_pair_leq)) ) ) ).

% pair_leqI2
tff(fact_7343_pair__leq__def,axiom,
    fun_pair_leq = aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),sup_sup(set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),fun_pair_less),id2(product_prod(nat,nat))) ).

% pair_leq_def
tff(fact_7344_smin__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] :
      ( ( X6 != bot_bot(set(product_prod(nat,nat))) )
     => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_strict)) ) ).

% smin_emptyI
tff(fact_7345_min__strict__def,axiom,
    fun_min_strict = min_ext(product_prod(nat,nat),fun_pair_less) ).

% min_strict_def
tff(fact_7346_pair__leqI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T5: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T5))),fun_pair_leq)) ) ).

% pair_leqI1
tff(fact_7347_wmax__insertI,axiom,
    ! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),Y),YS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_max_weak))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert(product_prod(nat,nat)),X),XS)),YS)),fun_max_weak)) ) ) ) ).

% wmax_insertI
tff(fact_7348_wmin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),X),XS))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_weak))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert(product_prod(nat,nat)),Y),YS))),fun_min_weak)) ) ) ) ).

% wmin_insertI
tff(fact_7349_wmin__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] : pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_weak)) ).

% wmin_emptyI
tff(fact_7350_wmax__emptyI,axiom,
    ! [X6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),X6))
     => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),X6)),fun_max_weak)) ) ).

% wmax_emptyI
tff(fact_7351_min__rpair__set,axiom,
    fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_min_strict),fun_min_weak)) ).

% min_rpair_set
tff(fact_7352_min__weak__def,axiom,
    fun_min_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),min_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),insert(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% min_weak_def
tff(fact_7353_max__weak__def,axiom,
    fun_max_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),max_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),insert(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% max_weak_def
tff(fact_7354_smax__insertI,axiom,
    ! [Y: product_prod(nat,nat),Y6: set(product_prod(nat,nat)),X: product_prod(nat,nat),X6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),Y),Y6))
     => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_less))
       => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),Y6)),fun_max_strict))
         => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),insert(product_prod(nat,nat)),X),X6)),Y6)),fun_max_strict)) ) ) ) ).

% smax_insertI
tff(fact_7355_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,B2)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_7356_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),euclid6346220572633701492n_size(A,B2)))
        <=> ( B2 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_7357_max__strict__def,axiom,
    fun_max_strict = max_ext(product_prod(nat,nat),fun_pair_less) ).

% max_strict_def
tff(fact_7358_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ).

% size_mult_mono
tff(fact_7359_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))) ) ) ).

% size_mult_mono'
tff(fact_7360_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( ( B2 != zero_zero(A) )
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_7361_max__rpair__set,axiom,
    fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_max_strict),fun_max_weak)) ).

% max_rpair_set
tff(fact_7362_dvd__imp__size__le,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ( B2 != zero_zero(A) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ).

% dvd_imp_size_le
tff(fact_7363_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,modulo_modulo(A,A3,B2))),euclid6346220572633701492n_size(A,B2))) ) ) ).

% mod_size_less
tff(fact_7364_smax__emptyI,axiom,
    ! [Y6: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),Y6))
     => ( ( Y6 != bot_bot(set(product_prod(nat,nat))) )
       => pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y6)),fun_max_strict)) ) ) ).

% smax_emptyI
tff(fact_7365_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( ( B2 != zero_zero(A) )
           => ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
             => ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ) )
         => ( ( ( B2 != zero_zero(A) )
             => ! [Q4: A,R: A] :
                  ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),euclid6346220572633701492n_size(A,B2)))
                   => ( ( R != zero_zero(A) )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q4 )
                       => ( ( modulo_modulo(A,A3,B2) = R )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q4),B2)),R) ) ) ) ) ) ) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_7366_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R3: A,Q5: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B2)),R3) = A3 )
               => ( modulo_modulo(A,A3,B2) = R3 ) ) ) ) ) ) ).

% mod_eqI
tff(fact_7367_division__segment__int__def,axiom,
    ! [K2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
       => ( euclid7384307370059645450egment(int,K2) = one_one(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
       => ( euclid7384307370059645450egment(int,K2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% division_segment_int_def
tff(fact_7368_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] :
          ( ( euclid7384307370059645450egment(A,A3) = euclid7384307370059645450egment(A,B2) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2)))
              | ( B2 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_7369_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R3: A,Q5: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B2)),R3) = A3 )
               => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q5 ) ) ) ) ) ) ).

% div_eqI
tff(fact_7370_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R3: A,Q5: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
             => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B2)),R3)),B2) = Q5 ) ) ) ) ) ).

% div_bounded
tff(fact_7371_smsI,axiom,
    ! [A6: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat)),Z5: multiset(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A6)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B5))),fun_max_strict))
     => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A6)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),B5))),ms_strict)) ) ).

% smsI
tff(fact_7372_wmsI,axiom,
    ! [A6: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat)),Z5: multiset(product_prod(nat,nat))] :
      ( ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A6)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B5))),fun_max_strict))
        | ( ( A6 = zero_zero(multiset(product_prod(nat,nat))) )
          & ( B5 = zero_zero(multiset(product_prod(nat,nat))) ) ) )
     => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A6)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),B5))),ms_weak)) ) ).

% wmsI
tff(fact_7373_ms__weak__def,axiom,
    ms_weak = aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),fun(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),sup_sup(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),ms_strict),id2(multiset(product_prod(nat,nat)))) ).

% ms_weak_def
tff(fact_7374_ms__weakI1,axiom,
    ! [Z5: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat)),A6: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z5,Z10)
     => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A6)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B5))),fun_max_strict))
       => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A6)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),B5))),ms_weak)) ) ) ).

% ms_weakI1
tff(fact_7375_ms__strictI,axiom,
    ! [Z5: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat)),A6: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z5,Z10)
     => ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A6)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B5))),fun_max_strict))
       => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),A6)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),B5))),ms_strict)) ) ) ).

% ms_strictI
tff(fact_7376_pw__leq_Ocases,axiom,
    ! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
      ( pw_leq(A1,A22)
     => ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
         => ( A22 != zero_zero(multiset(product_prod(nat,nat))) ) )
       => ~ ! [X3: product_prod(nat,nat),Y3: product_prod(nat,nat),X10: multiset(product_prod(nat,nat))] :
              ( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X3),zero_zero(multiset(product_prod(nat,nat))))),X10) )
             => ! [Y10: multiset(product_prod(nat,nat))] :
                  ( ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y3),zero_zero(multiset(product_prod(nat,nat))))),Y10) )
                 => ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X3),Y3)),fun_pair_leq))
                   => ~ pw_leq(X10,Y10) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_7377_pw__leq_Osimps,axiom,
    ! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
      ( pw_leq(A1,A22)
    <=> ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
          & ( A22 = zero_zero(multiset(product_prod(nat,nat))) ) )
        | ? [X4: product_prod(nat,nat),Y4: product_prod(nat,nat),X14: multiset(product_prod(nat,nat)),Y9: multiset(product_prod(nat,nat))] :
            ( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X4),zero_zero(multiset(product_prod(nat,nat))))),X14) )
            & ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y4),zero_zero(multiset(product_prod(nat,nat))))),Y9) )
            & pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X4),Y4)),fun_pair_leq))
            & pw_leq(X14,Y9) ) ) ) ).

% pw_leq.simps
tff(fact_7378_pw__leq__step,axiom,
    ! [X: product_prod(nat,nat),Y: product_prod(nat,nat),X6: multiset(product_prod(nat,nat)),Y6: multiset(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X),Y)),fun_pair_leq))
     => ( pw_leq(X6,Y6)
       => pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X),zero_zero(multiset(product_prod(nat,nat))))),X6),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y),zero_zero(multiset(product_prod(nat,nat))))),Y6)) ) ) ).

% pw_leq_step
tff(fact_7379_ms__weakI2,axiom,
    ! [Z5: multiset(product_prod(nat,nat)),Z10: multiset(product_prod(nat,nat))] :
      ( pw_leq(Z5,Z10)
     => pp(aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),bool,member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z5),zero_zero(multiset(product_prod(nat,nat))))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z10),zero_zero(multiset(product_prod(nat,nat)))))),ms_weak)) ) ).

% ms_weakI2
tff(fact_7380_pw__leq__split,axiom,
    ! [X6: multiset(product_prod(nat,nat)),Y6: multiset(product_prod(nat,nat))] :
      ( pw_leq(X6,Y6)
     => ? [A11: multiset(product_prod(nat,nat)),B8: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat))] :
          ( ( X6 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),A11),Z8) )
          & ( Y6 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),B8),Z8) )
          & ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),A11)),aa(multiset(product_prod(nat,nat)),set(product_prod(nat,nat)),set_mset(product_prod(nat,nat)),B8))),fun_max_strict))
            | ( ( B8 = zero_zero(multiset(product_prod(nat,nat))) )
              & ( A11 = zero_zero(multiset(product_prod(nat,nat))) ) ) ) ) ) ).

% pw_leq_split
tff(fact_7381_coinduct3__lemma,axiom,
    ! [A: $tType,X6: set(A),F3: fun(set(A),set(A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_arc(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F3)))))
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_arc(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F3))),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_arc(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F3))))) ) ) ).

% coinduct3_lemma
tff(fact_7382_def__coinduct3,axiom,
    ! [A: $tType,A6: set(A),F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( ( A6 = complete_lattice_gfp(set(A),F3) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( pp(aa(set(A),bool,member(A,A3),X6))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_ard(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),A6),F3),X6)))))
           => pp(aa(set(A),bool,member(A,A3),A6)) ) ) ) ) ).

% def_coinduct3
tff(fact_7383_def__Collect__coinduct,axiom,
    ! [A: $tType,A6: set(A),P2: fun(set(A),fun(A,bool)),A3: A,X6: set(A)] :
      ( ( A6 = complete_lattice_gfp(set(A),aTP_Lamp_are(fun(set(A),fun(A,bool)),fun(set(A),set(A)),P2)) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),aTP_Lamp_are(fun(set(A),fun(A,bool)),fun(set(A),set(A)),P2)))
       => ( pp(aa(set(A),bool,member(A,A3),X6))
         => ( ! [Z3: A] :
                ( pp(aa(set(A),bool,member(A,Z3),X6))
               => pp(aa(A,bool,aa(set(A),fun(A,bool),P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),A6)),Z3)) )
           => pp(aa(set(A),bool,member(A,A3),A6)) ) ) ) ) ).

% def_Collect_coinduct
tff(fact_7384_gfp__fun__UnI2,axiom,
    ! [A: $tType,F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
     => ( pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3)))
       => pp(aa(set(A),bool,member(A,A3),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),complete_lattice_gfp(set(A),F3))))) ) ) ).

% gfp_fun_UnI2
tff(fact_7385_gfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T5: A] : complete_lattice_gfp(A,aTP_Lamp_afz(A,fun(A,A),T5)) = T5 ) ).

% gfp_const
tff(fact_7386_gfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G3: fun(A,B),F3: fun(B,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),G3))
         => ( pp(aa(fun(B,A),bool,order_mono(B,A),F3))
           => ( aa(A,B,G3,complete_lattice_gfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_agc(fun(A,B),fun(fun(B,A),fun(A,A)),G3),F3))) = complete_lattice_gfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_agd(fun(A,B),fun(fun(B,A),fun(B,B)),G3),F3)) ) ) ) ) ).

% gfp_rolling
tff(fact_7387_gfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] : complete_lattice_gfp(A,F3) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_arf(fun(A,A),fun(A,bool),F3))) ) ).

% gfp_def
tff(fact_7388_gfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),X: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F4))
         => ( ( aa(A,A,F4,X) = X )
           => ( ! [Z3: A] :
                  ( ( aa(A,A,F4,Z3) = Z3 )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X)) )
             => ( complete_lattice_gfp(A,F4) = X ) ) ) ) ) ).

% gfp_eqI
tff(fact_7389_gfp__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),X6: A] :
          ( ! [U3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U3),aa(A,A,F3,U3)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U3),X6)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),X6)) ) ) ).

% gfp_least
tff(fact_7390_gfp__upperbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F3: fun(A,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,X6)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F3))) ) ) ).

% gfp_upperbound
tff(fact_7391_gfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),G3: fun(A,A)] :
          ( ! [Z8: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,Z8)),aa(A,A,G3,Z8)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),complete_lattice_gfp(A,G3))) ) ) ).

% gfp_mono
tff(fact_7392_weak__coinduct,axiom,
    ! [A: $tType,A3: A,X6: set(A),F3: fun(set(A),set(A))] :
      ( pp(aa(set(A),bool,member(A,A3),X6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,X6)))
       => pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3))) ) ) ).

% weak_coinduct
tff(fact_7393_gfp__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,fun(A,A))] :
          ( ! [X3: A,Y3: A,W: A,Z3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),W)),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3))) ) )
         => ( complete_lattice_gfp(A,aTP_Lamp_arg(fun(A,fun(A,A)),fun(A,A),F3)) = complete_lattice_gfp(A,aTP_Lamp_agb(fun(A,fun(A,A)),fun(A,A),F3)) ) ) ) ).

% gfp_gfp
tff(fact_7394_weak__coinduct__image,axiom,
    ! [A: $tType,B: $tType,A3: A,X6: set(A),G3: fun(A,B),F3: fun(set(B),set(B))] :
      ( pp(aa(set(A),bool,member(A,A3),X6))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G3),X6)),aa(set(B),set(B),F3,aa(set(A),set(B),image2(A,B,G3),X6))))
       => pp(aa(set(B),bool,member(B,aa(A,B,G3,A3)),complete_lattice_gfp(set(B),F3))) ) ) ).

% weak_coinduct_image
tff(fact_7395_coinduct__set,axiom,
    ! [A: $tType,F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
     => ( pp(aa(set(A),bool,member(A,A3),X6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),complete_lattice_gfp(set(A),F3)))))
         => pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3))) ) ) ) ).

% coinduct_set
tff(fact_7396_def__coinduct__set,axiom,
    ! [A: $tType,A6: set(A),F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( ( A6 = complete_lattice_gfp(set(A),F3) )
     => ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
       => ( pp(aa(set(A),bool,member(A,A3),X6))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X6),A6))))
           => pp(aa(set(A),bool,member(A,A3),A6)) ) ) ) ) ).

% def_coinduct_set
tff(fact_7397_coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),X6: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3)))))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F3))) ) ) ) ).

% coinduct
tff(fact_7398_def__coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: A,F3: fun(A,A),X6: A] :
          ( ( A6 = complete_lattice_gfp(A,F3) )
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),A6))))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),A6)) ) ) ) ) ).

% def_coinduct
tff(fact_7399_coinduct__lemma,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F3: fun(A,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3)))))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3))),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3))))) ) ) ) ).

% coinduct_lemma
tff(fact_7400_gfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: fun(A,bool)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ! [S6: A] :
                ( pp(aa(A,bool,P2,S6))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),S6))
                 => pp(aa(A,bool,P2,aa(A,A,F3,S6))) ) )
           => ( ! [M11: set(A)] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),M11))
                     => pp(aa(A,bool,P2,X5)) )
                 => pp(aa(A,bool,P2,aa(set(A),A,complete_Inf_Inf(A),M11))) )
             => pp(aa(A,bool,P2,complete_lattice_gfp(A,F3))) ) ) ) ) ).

% gfp_ordinal_induct
tff(fact_7401_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3)) = complete_lattice_gfp(A,F3) ) ) ) ).

% gfp_funpow
tff(fact_7402_lfp__le__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),complete_lattice_gfp(A,F3))) ) ) ).

% lfp_le_gfp
tff(fact_7403_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K2)),F3),top_top(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),top_top(A)) )
           => ( complete_lattice_gfp(A,F3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_7404_coinduct3,axiom,
    ! [A: $tType,F3: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( pp(aa(fun(set(A),set(A)),bool,order_mono(set(A),set(A)),F3))
     => ( pp(aa(set(A),bool,member(A,A3),X6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),F3,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_arh(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),F3),X6)))))
         => pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F3))) ) ) ) ).

% coinduct3
tff(fact_7405_AboveS__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] : order_AboveS(A,R3,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ari(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R3),A6)) ).

% AboveS_def
tff(fact_7406_right__total__relcompp__transfer,axiom,
    ! [A: $tType,E: $tType,C: $tType,F2: $tType,B: $tType,D: $tType,B5: fun(A,fun(B,bool)),A6: fun(C,fun(D,bool)),C4: fun(E,fun(F2,bool))] :
      ( right_total(A,B,B5)
     => pp(aa(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool)))),bool,aa(fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),fun(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool)))),bool),bNF_rel_fun(fun(C,fun(A,bool)),fun(D,fun(B,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool))),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool))),bNF_rel_fun(C,D,fun(A,bool),fun(B,bool),A6,bNF_rel_fun(A,B,bool,bool,B5,fequal(bool))),bNF_rel_fun(fun(A,fun(E,bool)),fun(B,fun(F2,bool)),fun(C,fun(E,bool)),fun(D,fun(F2,bool)),bNF_rel_fun(A,B,fun(E,bool),fun(F2,bool),B5,bNF_rel_fun(E,F2,bool,bool,C4,fequal(bool))),bNF_rel_fun(C,D,fun(E,bool),fun(F2,bool),A6,bNF_rel_fun(E,F2,bool,bool,C4,fequal(bool))))),aTP_Lamp_arj(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),B5)),relcompp(D,B,F2))) ) ).

% right_total_relcompp_transfer
tff(fact_7407_relcompp__distrib2,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: fun(A,fun(C,bool)),T8: fun(A,fun(C,bool)),R2: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(fun(A,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(C,bool)),fun(fun(A,fun(C,bool)),fun(A,fun(C,bool))),sup_sup(fun(A,fun(C,bool))),S),T8)),R2) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),S),R2)),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),T8),R2)) ).

% relcompp_distrib2
tff(fact_7408_relcompp__distrib,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(A,fun(C,bool)),S: fun(C,fun(B,bool)),T8: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),aa(fun(C,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(fun(C,fun(B,bool)),fun(C,fun(B,bool))),sup_sup(fun(C,fun(B,bool))),S),T8)) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),S)),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),T8)) ).

% relcompp_distrib
tff(fact_7409_relpowp__add,axiom,
    ! [A: $tType,M: nat,N: nat,P2: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),P2) = aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M),P2)),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2)) ).

% relpowp_add
tff(fact_7410_relcompp__relcomp__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(B,C)),X5: A,Xa: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),aTP_Lamp_ark(set(product_prod(B,C)),fun(B,fun(C,bool)),S2)),X5),Xa))
    <=> pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X5),Xa)),relcomp(A,B,C,R3,S2))) ) ).

% relcompp_relcomp_eq
tff(fact_7411_pos__fun__distr,axiom,
    ! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F2: $tType,R2: fun(A,fun(E,bool)),S: fun(B,fun(F2,bool)),R8: fun(E,fun(C,bool)),S4: fun(F2,fun(D,bool))] : pp(aa(fun(fun(A,B),fun(fun(C,D),bool)),bool,aa(fun(fun(A,B),fun(fun(C,D),bool)),fun(fun(fun(A,B),fun(fun(C,D),bool)),bool),ord_less_eq(fun(fun(A,B),fun(fun(C,D),bool))),aa(fun(fun(E,F2),fun(fun(C,D),bool)),fun(fun(A,B),fun(fun(C,D),bool)),aa(fun(fun(A,B),fun(fun(E,F2),bool)),fun(fun(fun(E,F2),fun(fun(C,D),bool)),fun(fun(A,B),fun(fun(C,D),bool))),relcompp(fun(A,B),fun(E,F2),fun(C,D)),bNF_rel_fun(A,E,B,F2,R2,S)),bNF_rel_fun(E,C,F2,D,R8,S4))),bNF_rel_fun(A,C,B,D,aa(fun(E,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(E,bool)),fun(fun(E,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,E,C),R2),R8),aa(fun(F2,fun(D,bool)),fun(B,fun(D,bool)),aa(fun(B,fun(F2,bool)),fun(fun(F2,fun(D,bool)),fun(B,fun(D,bool))),relcompp(B,F2,D),S),S4)))) ).

% pos_fun_distr
tff(fact_7412_relcompp__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,R6: fun(A,fun(B,bool)),R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool)),S2: fun(B,fun(C,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R6),R3))
     => ( pp(aa(fun(B,fun(C,bool)),bool,aa(fun(B,fun(C,bool)),fun(fun(B,fun(C,bool)),bool),ord_less_eq(fun(B,fun(C,bool))),S3),S2))
       => pp(aa(fun(A,fun(C,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(A,fun(C,bool)),bool),ord_less_eq(fun(A,fun(C,bool))),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R6),S3)),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S2))) ) ) ).

% relcompp_mono
tff(fact_7413_leq__OOI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( ( R2 = fequal(A) )
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R2),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R2),R2))) ) ).

% leq_OOI
tff(fact_7414_relcompp__SUP__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S2: fun(A,fun(C,bool)),R3: fun(D,fun(C,fun(B,bool))),I5: set(D)] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),S2),aa(set(fun(C,fun(B,bool))),fun(C,fun(B,bool)),complete_Sup_Sup(fun(C,fun(B,bool))),aa(set(D),set(fun(C,fun(B,bool))),image2(D,fun(C,fun(B,bool)),R3),I5))) = aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(D),set(fun(A,fun(B,bool))),image2(D,fun(A,fun(B,bool)),aa(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool))),aTP_Lamp_arl(fun(A,fun(C,bool)),fun(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool)))),S2),R3)),I5)) ).

% relcompp_SUP_distrib
tff(fact_7415_relcompp__SUP__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R3: fun(D,fun(A,fun(C,bool))),I5: set(D),S2: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(set(fun(A,fun(C,bool))),fun(A,fun(C,bool)),complete_Sup_Sup(fun(A,fun(C,bool))),aa(set(D),set(fun(A,fun(C,bool))),image2(D,fun(A,fun(C,bool)),R3),I5))),S2) = aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(D),set(fun(A,fun(B,bool))),image2(D,fun(A,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool))),aTP_Lamp_arm(fun(D,fun(A,fun(C,bool))),fun(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool)))),R3),S2)),I5)) ).

% relcompp_SUP_distrib2
tff(fact_7416_nchotomy__relcomppE,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: fun(B,A),R3: fun(C,fun(A,bool)),S2: fun(A,fun(D,bool)),A3: C,C3: D] :
      ( ! [Y3: A] :
        ? [X5: B] : Y3 = aa(B,A,F3,X5)
     => ( pp(aa(D,bool,aa(C,fun(D,bool),aa(fun(A,fun(D,bool)),fun(C,fun(D,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(D,bool)),fun(C,fun(D,bool))),relcompp(C,A,D),R3),S2),A3),C3))
       => ~ ! [B4: B] :
              ( pp(aa(A,bool,aa(C,fun(A,bool),R3,A3),aa(B,A,F3,B4)))
             => ~ pp(aa(D,bool,aa(A,fun(D,bool),S2,aa(B,A,F3,B4)),C3)) ) ) ) ).

% nchotomy_relcomppE
tff(fact_7417_pcr__Domainp,axiom,
    ! [C: $tType,B: $tType,A: $tType,B5: fun(A,fun(B,bool)),P2: fun(A,bool),A6: fun(C,fun(A,bool))] :
      ( ( aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),B5) = P2 )
     => ! [X5: C] :
          ( pp(aa(C,bool,aa(fun(C,fun(B,bool)),fun(C,bool),domainp(C,B),aa(fun(A,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,A,B),A6),B5)),X5))
        <=> ? [Y4: A] :
              ( pp(aa(A,bool,aa(C,fun(A,bool),A6,X5),Y4))
              & pp(aa(A,bool,P2,Y4)) ) ) ) ).

% pcr_Domainp
tff(fact_7418_OO__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: fun(A,fun(C,bool)),S: fun(C,fun(B,bool)),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),S),X5),Xa))
    <=> ? [Y4: C] :
          ( pp(aa(C,bool,aa(A,fun(C,bool),R2,X5),Y4))
          & pp(aa(B,bool,aa(C,fun(B,bool),S,Y4),Xa)) ) ) ).

% OO_def
tff(fact_7419_relcomp__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,X5: set(product_prod(A,B)),Xa: set(product_prod(B,C))] : relcomp(A,B,C,X5,Xa) = aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)),aTP_Lamp_ark(set(product_prod(B,C)),fun(B,fun(C,bool)),Xa)))) ).

% relcomp_def
tff(fact_7420_AboveS__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_AboveS(A,R3,A6)),field2(A,R3))) ).

% AboveS_Field
tff(fact_7421_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( ( order_AboveS(A,R3,B5) != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,member(A,B2),B5))
           => ( ( bNF_Wellorder_wo_suc(A,R3,B5) != B2 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R3,B5))),R3)) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_7422_wo__rel_Osuc__AboveS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( ( order_AboveS(A,R3,B5) != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_Wellorder_wo_suc(A,R3,B5)),order_AboveS(A,R3,B5))) ) ) ) ).

% wo_rel.suc_AboveS
tff(fact_7423_wo__rel_Osuc__least__AboveS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R3,B5)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R3,B5)),A3)),R3)) ) ) ).

% wo_rel.suc_least_AboveS
tff(fact_7424_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R3,B5)))
         => ( ! [A8: A] :
                ( pp(aa(set(A),bool,member(A,A8),order_AboveS(A,R3,B5)))
               => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A8)),R3)) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R3,B5) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_7425_wo__rel_Osuc__inField,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R3)))
       => ( ( order_AboveS(A,R3,B5) != bot_bot(set(A)) )
         => pp(aa(set(A),bool,member(A,bNF_Wellorder_wo_suc(A,R3,B5)),field2(A,R3))) ) ) ) ).

% wo_rel.suc_inField
tff(fact_7426_neg__fun__distr1,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,E: $tType,F2: $tType,R2: fun(A,fun(B,bool)),R8: fun(B,fun(C,bool)),S: fun(D,fun(F2,bool)),S4: fun(F2,fun(E,bool))] :
      ( left_unique(A,B,R2)
     => ( right_total(A,B,R2)
       => ( right_unique(B,C,R8)
         => ( left_total(B,C,R8)
           => pp(aa(fun(fun(A,D),fun(fun(C,E),bool)),bool,aa(fun(fun(A,D),fun(fun(C,E),bool)),fun(fun(fun(A,D),fun(fun(C,E),bool)),bool),ord_less_eq(fun(fun(A,D),fun(fun(C,E),bool))),bNF_rel_fun(A,C,D,E,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R2),R8),aa(fun(F2,fun(E,bool)),fun(D,fun(E,bool)),aa(fun(D,fun(F2,bool)),fun(fun(F2,fun(E,bool)),fun(D,fun(E,bool))),relcompp(D,F2,E),S),S4))),aa(fun(fun(B,F2),fun(fun(C,E),bool)),fun(fun(A,D),fun(fun(C,E),bool)),aa(fun(fun(A,D),fun(fun(B,F2),bool)),fun(fun(fun(B,F2),fun(fun(C,E),bool)),fun(fun(A,D),fun(fun(C,E),bool))),relcompp(fun(A,D),fun(B,F2),fun(C,E)),bNF_rel_fun(A,B,D,F2,R2,S)),bNF_rel_fun(B,C,F2,E,R8,S4)))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_7427_neg__fun__distr2,axiom,
    ! [F2: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R8: fun(A,fun(B,bool)),S4: fun(C,fun(D,bool)),R2: fun(E,fun(A,bool)),S: fun(F2,fun(C,bool))] :
      ( right_unique(A,B,R8)
     => ( left_total(A,B,R8)
       => ( left_unique(C,D,S4)
         => ( right_total(C,D,S4)
           => pp(aa(fun(fun(E,F2),fun(fun(B,D),bool)),bool,aa(fun(fun(E,F2),fun(fun(B,D),bool)),fun(fun(fun(E,F2),fun(fun(B,D),bool)),bool),ord_less_eq(fun(fun(E,F2),fun(fun(B,D),bool))),bNF_rel_fun(E,B,F2,D,aa(fun(A,fun(B,bool)),fun(E,fun(B,bool)),aa(fun(E,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(E,fun(B,bool))),relcompp(E,A,B),R2),R8),aa(fun(C,fun(D,bool)),fun(F2,fun(D,bool)),aa(fun(F2,fun(C,bool)),fun(fun(C,fun(D,bool)),fun(F2,fun(D,bool))),relcompp(F2,C,D),S),S4))),aa(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(E,F2),fun(fun(B,D),bool)),aa(fun(fun(E,F2),fun(fun(A,C),bool)),fun(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(E,F2),fun(fun(B,D),bool))),relcompp(fun(E,F2),fun(A,C),fun(B,D)),bNF_rel_fun(E,A,F2,C,R2,S)),bNF_rel_fun(A,B,C,D,R8,S4)))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_7428_typedef__right__unique,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A6: set(A),T8: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A6)
     => ( ! [X3: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X3),Xa4))
          <=> ( X3 = aa(B,A,Rep,Xa4) ) )
       => right_unique(A,B,T8) ) ) ).

% typedef_right_unique
tff(fact_7429_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A6)
       => ( ( order_AboveS(A,R3,A6) != bot_bot(set(A)) )
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R3,A6))),R3))
           => ( ( B2 != bNF_Wellorder_wo_suc(A,R3,A6) )
             => pp(aa(set(A),bool,member(A,B2),A6)) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_7430_cSUP__UNION,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A6: set(C),B5: fun(C,set(D)),F3: fun(D,B)] :
          ( ( A6 != bot_bot(set(C)) )
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,member(C,X3),A6))
               => ( aa(C,set(D),B5,X3) != bot_bot(set(D)) ) )
           => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_arn(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B5),F3)),A6)))
             => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,F3),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B5),A6)))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_aro(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B5),F3)),A6)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_7431_bdd__above__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A6: set(A)] : condit941137186595557371_above(A,A6) ) ).

% bdd_above_top
tff(fact_7432_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),M6: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),M6)) )
         => condit941137186595557371_above(A,A6) ) ) ).

% bdd_above.I
tff(fact_7433_bdd__above__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit941137186595557371_above(A,bot_bot(set(A))) ) ).

% bdd_above_empty
tff(fact_7434_bdd__above__insert,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: A,A6: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6))
        <=> condit941137186595557371_above(A,A6) ) ) ).

% bdd_above_insert
tff(fact_7435_bdd__above__Un,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))
        <=> ( condit941137186595557371_above(A,A6)
            & condit941137186595557371_above(A,B5) ) ) ) ).

% bdd_above_Un
tff(fact_7436_bdd__above__Icc,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit941137186595557371_above(A,set_or1337092689740270186AtMost(A,A3,B2)) ) ).

% bdd_above_Icc
tff(fact_7437_bdd__above__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit941137186595557371_above(A,set_or7035219750837199246ssThan(A,A3,B2)) ) ).

% bdd_above_Ico
tff(fact_7438_bdd__above__Iic,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A] : condit941137186595557371_above(A,aa(A,set(A),set_ord_atMost(A),B2)) ) ).

% bdd_above_Iic
tff(fact_7439_bdd__above__Iio,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A] : condit941137186595557371_above(A,aa(A,set(A),set_ord_lessThan(A),B2)) ) ).

% bdd_above_Iio
tff(fact_7440_bdd__above__Ioc,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit941137186595557371_above(A,set_or3652927894154168847AtMost(A,A3,B2)) ) ).

% bdd_above_Ioc
tff(fact_7441_bdd__above__Ioo,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit941137186595557371_above(A,set_or5935395276787703475ssThan(A,A3,B2)) ) ).

% bdd_above_Ioo
tff(fact_7442_bdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_arp(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A6))
        <=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
            & condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G3),A6)) ) ) ) ).

% bdd_above_image_sup
tff(fact_7443_bdd__above__UN,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [I5: set(B),A6: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( condit941137186595557371_above(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5)))
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),I5))
               => condit941137186595557371_above(A,aa(B,set(A),A6,X4)) ) ) ) ) ).

% bdd_above_UN
tff(fact_7444_cSUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: B,A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,X),A6))
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ) ).

% cSUP_upper
tff(fact_7445_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),X: B,U: A] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
         => ( pp(aa(set(B),bool,member(B,X),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ) ) ).

% cSUP_upper2
tff(fact_7446_bdd__above__image__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( condit941137186595557371_above(A,A6)
           => condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A6)) ) ) ) ).

% bdd_above_image_mono
tff(fact_7447_bdd__above__Int1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( condit941137186595557371_above(A,A6)
         => condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ) ) ).

% bdd_above_Int1
tff(fact_7448_bdd__above__Int2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: set(A),A6: set(A)] :
          ( condit941137186595557371_above(A,B5)
         => condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ) ) ).

% bdd_above_Int2
tff(fact_7449_bdd__above__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: set(A),A6: set(A)] :
          ( condit941137186595557371_above(A,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
           => condit941137186595557371_above(A,A6) ) ) ) ).

% bdd_above_mono
tff(fact_7450_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( condit941137186595557371_above(A,A6)
         => ~ ! [M11: A] :
                ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),M11)) ) ) ) ).

% bdd_above.E
tff(fact_7451_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( condit941137186595557371_above(A,A6)
        <=> ? [M13: A] :
            ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M13)) ) ) ) ).

% bdd_above.unfold
tff(fact_7452_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A),Y: A] :
          ( pp(aa(set(A),bool,member(A,X),X6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( condit941137186595557371_above(A,X6)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ) ).

% cSup_upper2
tff(fact_7453_cSup__upper,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A)] :
          ( pp(aa(set(A),bool,member(A,X),X6))
         => ( condit941137186595557371_above(A,X6)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% cSup_upper
tff(fact_7454_bdd__above_OI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A6: set(B),F3: fun(B,A),M6: A] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),M6)) )
         => condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6)) ) ) ).

% bdd_above.I2
tff(fact_7455_wo__rel_Oofilter__linord,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A6)
       => ( order_ofilter(A,R3,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A6)) ) ) ) ) ).

% wo_rel.ofilter_linord
tff(fact_7456_bdd__above__nat,axiom,
    ! [X6: set(nat)] :
      ( condit941137186595557371_above(nat,X6)
    <=> pp(aa(set(nat),bool,finite_finite2(nat),X6)) ) ).

% bdd_above_nat
tff(fact_7457_bdd__above__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => condit941137186595557371_above(A,A6) ) ) ).

% bdd_above_finite
tff(fact_7458_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B5: set(A),A6: set(A)] :
          ( ( B5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A6)
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,member(A,B4),B5))
                 => ? [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),X5)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B5)),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ) ).

% cSup_mono
tff(fact_7459_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A3))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),S))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A3)) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_7460_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X6)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),X6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X4)) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_7461_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),Y: A,I2: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),Y))
           => ( pp(aa(set(B),bool,member(B,I2),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I2)),Y)) ) ) ) ) ).

% cSUP_lessD
tff(fact_7462_cSUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),U))
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_7463_cSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),G3: fun(C,A),B5: set(C),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,G3),B5))
           => ( ! [N4: B] :
                  ( pp(aa(set(B),bool,member(B,N4),A6))
                 => ? [X5: C] :
                      ( pp(aa(set(C),bool,member(C,X5),B5))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N4)),aa(C,A,G3,X5))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G3),B5)))) ) ) ) ) ).

% cSUP_mono
tff(fact_7464_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B5)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ) ) ).

% cSup_subset_mono
tff(fact_7465_cSup__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X6)
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X6)) ) ) ) ) ).

% cSup_insert
tff(fact_7466_cSup__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( condit941137186595557371_above(A,X6)
         => ( ( ( X6 = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)) = A3 ) )
            & ( ( X6 != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X6)) ) ) ) ) ) ).

% cSup_insert_If
tff(fact_7467_cSup__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A6)
           => ( ( B5 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,B5)
               => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_7468_wo__rel_Oofilter__UNION,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A6: fun(B,set(A))] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),I5))
           => order_ofilter(A,R3,aa(B,set(A),A6,I3)) )
       => order_ofilter(A,R3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))) ) ) ).

% wo_rel.ofilter_UNION
tff(fact_7469_less__cSUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A6: set(B),F3: fun(B,A),A3: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))))
            <=> ? [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,X4))) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_7470_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G3),A6))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),A6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_arq(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A6)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_7471_cSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),G3: fun(B,A),B5: set(B),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G3),B5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G3),B5)))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_7472_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( condit941137186595557371_above(A,A6)
         => ( condit941137186595557371_above(A,B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) != bot_bot(set(A)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),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)),A6),B5))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B5)))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_7473_cSUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),A3: B] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A6))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F3,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))) ) ) ) ) ).

% cSUP_insert
tff(fact_7474_cSUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),B5: set(B)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( ( B5 != bot_bot(set(B)) )
             => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),B5))
               => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),B5))) ) ) ) ) ) ) ).

% cSUP_union
tff(fact_7475_wo__rel_Oofilter__AboveS__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A6)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),order_AboveS(A,R3,A6)) = field2(A,R3) ) ) ) ).

% wo_rel.ofilter_AboveS_Field
tff(fact_7476_bdd__above__multiset__imp__bdd__above__count,axiom,
    ! [A: $tType,A6: set(multiset(A)),X: A] :
      ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),A6))
     => condit941137186595557371_above(nat,aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_akn(A,fun(multiset(A),nat),X)),A6)) ) ).

% bdd_above_multiset_imp_bdd_above_count
tff(fact_7477_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( aa(set(A),A,complete_Sup_Sup(A),S) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_arr(set(A),fun(A,bool),S))) ) ) ) ) ).

% cSup_cInf
tff(fact_7478_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( condit941137186595557371_above(A,A6)
           => ( ( A6 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A6)))) ) ) ) ) ).

% mono_cSup
tff(fact_7479_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A6),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ars(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6)),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A6),I5))))) ) ) ) ) ).

% mono_cSUP
tff(fact_7480_ofilterIncl__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R3) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_art(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R3))) ).

% ofilterIncl_def
tff(fact_7481_wo__rel_Oofilter__under__UNION,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A6)
       => ( A6 = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),order_under(A,R3)),A6)) ) ) ) ).

% wo_rel.ofilter_under_UNION
tff(fact_7482_under__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : aa(A,set(A),order_under(A,R3),A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_ake(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),A3)) ).

% under_def
tff(fact_7483_under__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),A3)),field2(A,R3))) ).

% under_Field
tff(fact_7484_under__incr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),A3)),aa(A,set(A),order_under(A,R3),B2))) ) ) ).

% under_incr
tff(fact_7485_ofilter__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_ofilter(A,R3,A6)
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),field2(A,R3)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A6))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),X4)),A6)) ) ) ) ).

% ofilter_def
tff(fact_7486_wo__rel_Oofilter__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R3)
     => ( order_ofilter(A,R3,A6)
      <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),field2(A,R3)))
          & ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A6))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R3),X4)),A6)) ) ) ) ) ).

% wo_rel.ofilter_def
tff(fact_7487_bsqr__ofilter,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),D5: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R3),D5)
       => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less(set(product_prod(A,A))),D5),product_Sigma(A,A,field2(A,R3),aTP_Lamp_aiy(set(product_prod(A,A)),fun(A,set(A)),R3))))
         => ( ~ ? [A5: A] : field2(A,R3) = aa(A,set(A),order_under(A,R3),A5)
           => ? [A11: set(A)] :
                ( order_ofilter(A,R3,A11)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A11),field2(A,R3)))
                & pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),D5),product_Sigma(A,A,A11,aTP_Lamp_abo(set(A),fun(A,set(A)),A11)))) ) ) ) ) ) ).

% bsqr_ofilter
tff(fact_7488_Refl__under__underS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,field2(A,R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( aa(A,set(A),order_under(A,R3),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R3,A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_7489_underS__subset__under,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),aa(A,set(A),order_under(A,R3),A3))) ).

% underS_subset_under
tff(fact_7490_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [A: $tType,I2: A,R2: set(product_prod(A,A)),J2: A] :
      ( pp(aa(set(A),bool,member(A,I2),order_underS(A,R2,J2)))
     => pp(aa(set(A),bool,member(A,I2),field2(A,R2))) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_7491_underS__Field2,axiom,
    ! [A: $tType,A3: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R3,A3)),field2(A,R3))) ) ).

% underS_Field2
tff(fact_7492_Order__Relation_OunderS__Field,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),field2(A,R3))) ).

% Order_Relation.underS_Field
tff(fact_7493_underS__I,axiom,
    ! [A: $tType,I2: A,J2: A,R2: set(product_prod(A,A))] :
      ( ( I2 != J2 )
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2))
       => pp(aa(set(A),bool,member(A,I2),order_underS(A,R2,J2))) ) ) ).

% underS_I
tff(fact_7494_underS__E,axiom,
    ! [A: $tType,I2: A,R2: set(product_prod(A,A)),J2: A] :
      ( pp(aa(set(A),bool,member(A,I2),order_underS(A,R2,J2)))
     => ( ( I2 != J2 )
        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2)) ) ) ).

% underS_E
tff(fact_7495_underS__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] : order_underS(A,R3,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aru(set(product_prod(A,A)),fun(A,fun(A,bool)),R3),A3)) ).

% underS_def
tff(fact_7496_well__order__on__domain,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_well_order_on(A,A6,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( pp(aa(set(A),bool,member(A,A3),A6))
          & pp(aa(set(A),bool,member(A,B2),A6)) ) ) ) ).

% well_order_on_domain
tff(fact_7497_Well__order__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => order_well_order_on(A,field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% Well_order_Restr
tff(fact_7498_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_well_order_on(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_well_order_on
tff(fact_7499_underS__Field3,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( ( field2(A,R3) != bot_bot(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R3,A3)),field2(A,R3))) ) ).

% underS_Field3
tff(fact_7500_well__order__on__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),field2(A,R3)))
       => order_well_order_on(A,A6,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ) ).

% well_order_on_Restr
tff(fact_7501_Field__Restr__ofilter,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) = A6 ) ) ) ).

% Field_Restr_ofilter
tff(fact_7502_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_well_order_on(nat,field2(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),N)))),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_Well_order
tff(fact_7503_underS__incl__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R3),R3)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R3)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),order_underS(A,R3,B2)))
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ) ) ).

% underS_incl_iff
tff(fact_7504_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R3),R3)
     => ( order_well_order_on(A,field2(A,R3),R3)
      <=> ! [A16: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A16),field2(A,R3)))
           => ( ( A16 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),A16))
                  & ! [Xa3: A] :
                      ( pp(aa(set(A),bool,member(A,Xa3),A16))
                     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R3)) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_7505_ofilter__Restr__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
         => order_ofilter(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))),A6) ) ) ) ).

% ofilter_Restr_subset
tff(fact_7506_ofilter__Restr__Int,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => order_ofilter(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ) ) ).

% ofilter_Restr_Int
tff(fact_7507_bsqr__max2,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A1: A,A22: A,B1: A,B22: A] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( pp(aa(set(product_prod(product_prod(A,A),product_prod(A,A))),bool,member(product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B1),B22))),bNF_Wellorder_bsqr(A,R3)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R3,A1,A22)),bNF_We1388413361240627857o_max2(A,R3,B1,B22))),R3)) ) ) ).

% bsqr_max2
tff(fact_7508_ofilter__Restr__under,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),A3: A] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( pp(aa(set(A),bool,member(A,A3),A6))
         => ( aa(A,set(A),order_under(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),A3) = aa(A,set(A),order_under(A,R3),A3) ) ) ) ) ).

% ofilter_Restr_under
tff(fact_7509_UNION__inj__on__ofilter,axiom,
    ! [C: $tType,A: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A6: fun(B,set(A)),F3: fun(A,C)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( ! [I3: B] :
            ( pp(aa(set(B),bool,member(B,I3),I5))
           => order_ofilter(A,R3,aa(B,set(A),A6,I3)) )
       => ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),I5))
             => inj_on(A,C,F3,aa(B,set(A),A6,I3)) )
         => inj_on(A,C,F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5))) ) ) ) ).

% UNION_inj_on_ofilter
tff(fact_7510_ofilter__subset__embedS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( order_ofilter(A,R3,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
          <=> bNF_Wellorder_embedS(A,A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))),id(A)) ) ) ) ) ).

% ofilter_subset_embedS
tff(fact_7511_embedS__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( bNF_Wellorder_embedS(A,B,R3,R6,F3)
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F3),field2(A,R3))),field2(B,R6))) ) ) ).

% embedS_Field
tff(fact_7512_ofilter__subset__embedS__iso,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( order_ofilter(A,R3,B5)
         => ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
            <=> bNF_Wellorder_embedS(A,A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))),id(A)) )
            & ( ( A6 = B5 )
            <=> bNF_Wellorder_iso(A,A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))),id(A)) ) ) ) ) ) ).

% ofilter_subset_embedS_iso
tff(fact_7513_ofilter__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),field2(A,R3)))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),R3)),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).

% ofilter_ordLess
tff(fact_7514_iso__forward,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
     => ( bNF_Wellorder_iso(A,B,R3,R6,F3)
       => pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,X)),aa(A,B,F3,Y))),R6)) ) ) ).

% iso_forward
tff(fact_7515_iso__iff2,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( bNF_Wellorder_iso(A,B,R3,R6,F3)
    <=> ( bij_betw(A,B,F3,field2(A,R3),field2(B,R6))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),field2(A,R3)))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),field2(A,R3)))
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R3))
                <=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,X4)),aa(A,B,F3,Xa3))),R6)) ) ) ) ) ) ).

% iso_iff2
tff(fact_7516_ordLess__def,axiom,
    ! [A2: $tType,A: $tType] : bNF_We4044943003108391690rdLess(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_arv(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordLess_def
tff(fact_7517_underS__Restr__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( ( field2(A,R3) != bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,A3),aa(A,fun(A,set(A)),aTP_Lamp_arw(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),A3)))),R3)),bNF_We4044943003108391690rdLess(A,A))) ) ) ).

% underS_Restr_ordLess
tff(fact_7518_ofilter__subset__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( order_ofilter(A,R3,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B5))
          <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ) ).

% ofilter_subset_ordLess
tff(fact_7519_ordLess__iff__ordIso__Restr,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_well_order_on(B,field2(B,R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_We4044943003108391690rdLess(B,A)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),field2(A,R3)))
              & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_arw(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4))))),bNF_Wellorder_ordIso(B,A))) ) ) ) ) ).

% ordLess_iff_ordIso_Restr
tff(fact_7520_ordLeq__iff__ordLess__Restr,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_well_order_on(B,field2(B,R6),R6)
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),field2(A,R3)))
             => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_arw(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4)))),R6)),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).

% ordLeq_iff_ordLess_Restr
tff(fact_7521_internalize__ordLeq,axiom,
    ! [A: $tType,B: $tType,R6: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),R3)),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [P6: set(product_prod(B,B))] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),field2(B,P6)),field2(B,R3)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),P6)),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P6),R3)),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_ordLeq
tff(fact_7522_internalize__ordLess,axiom,
    ! [A: $tType,B: $tType,R6: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),R3)),bNF_We4044943003108391690rdLess(A,B)))
    <=> ? [P6: set(product_prod(B,B))] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),field2(B,P6)),field2(B,R3)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R6),P6)),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P6),R3)),bNF_We4044943003108391690rdLess(B,B))) ) ) ).

% internalize_ordLess
tff(fact_7523_ordIso__def,axiom,
    ! [A2: $tType,A: $tType] : bNF_Wellorder_ordIso(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_arx(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordIso_def
tff(fact_7524_ofilter__subset__ordLeq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( order_ofilter(A,R3,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
          <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5))))),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% ofilter_subset_ordLeq
tff(fact_7525_ofilter__embed,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
      <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),field2(A,R3)))
          & pp(aa(fun(A,A),bool,bNF_Wellorder_embed(A,A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))),R3),id(A))) ) ) ) ).

% ofilter_embed
tff(fact_7526_ofilter__subset__embed,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A),B5: set(A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_ofilter(A,R3,A6)
       => ( order_ofilter(A,R3,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
          <=> pp(aa(fun(A,A),bool,bNF_Wellorder_embed(A,A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B5,aTP_Lamp_abo(set(A),fun(A,set(A)),B5)))),id(A))) ) ) ) ) ).

% ofilter_subset_embed
tff(fact_7527_embed__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R3,R6),F3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),field2(A,R3))),field2(B,R6))) ) ).

% embed_Field
tff(fact_7528_iso__defs_I2_J,axiom,
    ! [A: $tType,A2: $tType,X5: set(product_prod(A,A)),Xa: set(product_prod(A2,A2)),Xb: fun(A,A2)] :
      ( bNF_Wellorder_iso(A,A2,X5,Xa,Xb)
    <=> ( pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,X5,Xa),Xb))
        & bij_betw(A,A2,Xb,field2(A,X5),field2(A2,Xa)) ) ) ).

% iso_defs(2)
tff(fact_7529_embedS__defs_I2_J,axiom,
    ! [A: $tType,A2: $tType,X5: set(product_prod(A,A)),Xa: set(product_prod(A2,A2)),Xb: fun(A,A2)] :
      ( bNF_Wellorder_embedS(A,A2,X5,Xa,Xb)
    <=> ( pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,X5,Xa),Xb))
        & ~ bij_betw(A,A2,Xb,field2(A,X5),field2(A2,Xa)) ) ) ).

% embedS_defs(2)
tff(fact_7530_embed__defs_I2_J,axiom,
    ! [A2: $tType,A: $tType,X5: set(product_prod(A,A)),Xa: set(product_prod(A2,A2)),Xb: fun(A,A2)] :
      ( pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,X5,Xa),Xb))
    <=> ! [Xc2: A] :
          ( pp(aa(set(A),bool,member(A,Xc2),field2(A,X5)))
         => bij_betw(A,A2,Xb,aa(A,set(A),order_under(A,X5),Xc2),aa(A2,set(A2),order_under(A2,Xa),aa(A,A2,Xb,Xc2))) ) ) ).

% embed_defs(2)
tff(fact_7531_embedS__iff,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(A,B)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R3,R6),F3))
       => ( bNF_Wellorder_embedS(A,B,R3,R6,F3)
        <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F3),field2(A,R3))),field2(B,R6))) ) ) ) ).

% embedS_iff
tff(fact_7532_ordLeq__def,axiom,
    ! [A2: $tType,A: $tType] : bNF_Wellorder_ordLeq(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_ary(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordLeq_def
tff(fact_7533_embed__implies__iso__Restr,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),F3: fun(B,A)] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( order_well_order_on(B,field2(B,R6),R6)
       => ( pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R6,R3),F3))
         => bNF_Wellorder_iso(B,A,R6,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F3),field2(B,R6)),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_arz(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),R6),F3))),F3) ) ) ) ).

% embed_implies_iso_Restr
tff(fact_7534_comp__set__bd__Union__o__collect,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: C,X6: set(fun(C,set(set(A)))),Hbd: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(fun(C,set(set(A)))),set(set(set(A))),image2(fun(C,set(set(A))),set(set(A)),aTP_Lamp_asa(C,fun(fun(C,set(set(A))),set(set(A))),X)),X6))))),Hbd)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A))),bNF_collect(C,set(A),X6)),X))),Hbd)),bNF_Wellorder_ordLeq(A,B))) ) ).

% comp_set_bd_Union_o_collect
tff(fact_7535_underS__incr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R3)
     => ( antisym(A,R3)
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R3,A3)),order_underS(A,R3,B2))) ) ) ) ).

% underS_incr
tff(fact_7536_antisym__reflcl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( antisym(A,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))),R3),id2(A)))
    <=> antisym(A,R3) ) ).

% antisym_reflcl
tff(fact_7537_card__of__Pow,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(set(A),pow(A,A6)))),bNF_We4044943003108391690rdLess(A,set(A)))) ).

% card_of_Pow
tff(fact_7538_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R3))),R6)),bNF_We4044943003108391690rdLess(A,B))) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_7539_antisym__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( antisym(A,R3)
    <=> ! [X4: A,Y4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4)),R3))
           => ( X4 = Y4 ) ) ) ) ).

% antisym_def
tff(fact_7540_antisymI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R3))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),R3))
           => ( X3 = Y3 ) ) )
     => antisym(A,R3) ) ).

% antisymI
tff(fact_7541_antisymD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A,B2: A] :
      ( antisym(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3))
         => ( A3 = B2 ) ) ) ) ).

% antisymD
tff(fact_7542_antisym__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => ( antisym(A,S2)
       => antisym(A,R3) ) ) ).

% antisym_subset
tff(fact_7543_Field__card__of,axiom,
    ! [A: $tType,A6: set(A)] : field2(A,bNF_Ca6860139660246222851ard_of(A,A6)) = A6 ).

% Field_card_of
tff(fact_7544_card__of__well__order__on,axiom,
    ! [A: $tType,A6: set(A)] : order_well_order_on(A,A6,bNF_Ca6860139660246222851ard_of(A,A6)) ).

% card_of_well_order_on
tff(fact_7545_card__of__Well__order,axiom,
    ! [A: $tType,A6: set(A)] : order_well_order_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(A,A6)) ).

% card_of_Well_order
tff(fact_7546_card__of__least,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,A6,R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_least
tff(fact_7547_card__of__Times2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ).

% card_of_Times2
tff(fact_7548_card__of__Times1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_ws(set(A),fun(B,set(A)),A6))))),bNF_Wellorder_ordLeq(B,product_prod(B,A)))) ) ).

% card_of_Times1
tff(fact_7549_card__of__Times__infinite__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B5 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_ws(set(A),fun(B,set(A)),A6))))),bNF_Wellorder_ordIso(A,product_prod(B,A)))) ) ) ) ).

% card_of_Times_infinite_simps(4)
tff(fact_7550_card__of__Times__infinite__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B5 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_ws(set(A),fun(B,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ).

% card_of_Times_infinite_simps(3)
tff(fact_7551_card__of__Times__infinite__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B5 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))),bNF_Wellorder_ordIso(A,product_prod(A,B)))) ) ) ) ).

% card_of_Times_infinite_simps(2)
tff(fact_7552_card__of__Times__infinite__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B5 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).

% card_of_Times_infinite_simps(1)
tff(fact_7553_card__of__Times__infinite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B5 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(A,B),A)))
            & pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_ws(set(A),fun(B,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ).

% card_of_Times_infinite
tff(fact_7554_card__of__empty,axiom,
    ! [B: $tType,A: $tType,A6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ).

% card_of_empty
tff(fact_7555_card__of__empty2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,B)))
     => ( A6 = bot_bot(set(A)) ) ) ).

% card_of_empty2
tff(fact_7556_card__of__empty3,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordLeq(A,B)))
     => ( A6 = bot_bot(set(A)) ) ) ).

% card_of_empty3
tff(fact_7557_card__of__empty__ordIso,axiom,
    ! [B: $tType,A: $tType] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,B))) ).

% card_of_empty_ordIso
tff(fact_7558_card__of__bool,axiom,
    ! [A: $tType,A1: A,A22: A] :
      ( ( A1 != A22 )
     => pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(bool,top_top(set(bool)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A22),bot_bot(set(A))))))),bNF_Wellorder_ordIso(bool,A))) ) ).

% card_of_bool
tff(fact_7559_card__of__mono1,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_mono1
tff(fact_7560_Func__Times__Range,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),bool,member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Ca6860139660246222851ard_of(fun(A,product_prod(B,C)),bNF_Wellorder_Func(A,product_prod(B,C),A6,product_Sigma(B,C,B5,aTP_Lamp_abq(set(C),fun(B,set(C)),C4))))),bNF_Ca6860139660246222851ard_of(product_prod(fun(A,B),fun(A,C)),product_Sigma(fun(A,B),fun(A,C),bNF_Wellorder_Func(A,B,A6,B5),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_asb(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),A6),C4))))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).

% Func_Times_Range
tff(fact_7561_card__of__Func__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),bool,member(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),fun(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),product_Pair(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),bNF_Ca6860139660246222851ard_of(fun(product_prod(A,B),C),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),C4))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,C)),bNF_Wellorder_Func(A,fun(B,C),A6,bNF_Wellorder_Func(B,C,B5,C4))))),bNF_Wellorder_ordIso(fun(product_prod(A,B),C),fun(A,fun(B,C))))) ).

% card_of_Func_Times
tff(fact_7562_card__of__UNION__Sigma,axiom,
    ! [B: $tType,A: $tType,A6: fun(B,set(A)),I5: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,I5,A6)))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ).

% card_of_UNION_Sigma
tff(fact_7563_card__of__Times3,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6))))),bNF_Wellorder_ordLeq(A,product_prod(A,A)))) ).

% card_of_Times3
tff(fact_7564_card__of__Sigma__mono1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B5: fun(A,set(C))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),I5))
         => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A6,X3))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),B5,X3)))),bNF_Wellorder_ordLeq(B,C))) )
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I5,A6))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,I5,B5)))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C)))) ) ).

% card_of_Sigma_mono1
tff(fact_7565_card__of__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,A6,aTP_Lamp_abr(set(C),fun(A,set(C)),C4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,B5,aTP_Lamp_abq(set(C),fun(B,set(C)),C4))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% card_of_Times_mono1
tff(fact_7566_card__of__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,C4,aTP_Lamp_asc(set(A),fun(C,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,C4,aTP_Lamp_asd(set(B),fun(C,set(B)),B5))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% card_of_Times_mono2
tff(fact_7567_card__of__Times__commute,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] : pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_ws(set(A),fun(B,set(A)),A6))))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).

% card_of_Times_commute
tff(fact_7568_card__of__Sigma__ordLeq__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,B5: set(A),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(B,A)))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),I5))
             => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(C,A))) )
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A6))),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ).

% card_of_Sigma_ordLeq_infinite
tff(fact_7569_card__of__Times__same__infinite,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(A,A),A))) ) ).

% card_of_Times_same_infinite
tff(fact_7570_card__of__ordIso,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ? [F11: fun(A,B)] : bij_betw(A,B,F11,A6,B5)
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_ordIso
tff(fact_7571_card__of__ordIsoI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B5: set(B)] :
      ( bij_betw(A,B,F3,A6,B5)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_ordIsoI
tff(fact_7572_ex__bij__betw,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ? [F: fun(B,A),B8: set(B)] : bij_betw(B,A,F,B8,A6) ) ).

% ex_bij_betw
tff(fact_7573_infinite__iff__card__of__nat,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
    <=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(nat,A))) ) ).

% infinite_iff_card_of_nat
tff(fact_7574_card__of__ordIso__finite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(A),bool,finite_finite2(A),A6))
      <=> pp(aa(set(B),bool,finite_finite2(B),B5)) ) ) ).

% card_of_ordIso_finite
tff(fact_7575_card__of__ordLeq__finite,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% card_of_ordLeq_finite
tff(fact_7576_card__of__ordLeq__infinite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
       => ~ pp(aa(set(B),bool,finite_finite2(B),B5)) ) ) ).

% card_of_ordLeq_infinite
tff(fact_7577_card__of__refl,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(A,A))) ).

% card_of_refl
tff(fact_7578_card__of__ordLeqI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A6)
     => ( ! [A5: A] :
            ( pp(aa(set(A),bool,member(A,A5),A6))
           => pp(aa(set(B),bool,member(B,aa(A,B,F3,A5)),B5)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B))) ) ) ).

% card_of_ordLeqI
tff(fact_7579_card__of__cong,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R3))),bNF_Ca6860139660246222851ard_of(B,field2(B,R6)))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_cong
tff(fact_7580_card__of__mono2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R3))),bNF_Ca6860139660246222851ard_of(B,field2(B,R6)))),bNF_Wellorder_ordLeq(A,B))) ) ).

% card_of_mono2
tff(fact_7581_card__of__image,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(B),set(A),image2(B,A,F3),A6))),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ).

% card_of_image
tff(fact_7582_card__of__Pow__Func,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),bNF_Ca6860139660246222851ard_of(set(A),pow(A,A6))),bNF_Ca6860139660246222851ard_of(fun(A,bool),bNF_Wellorder_Func(A,bool,A6,top_top(set(bool)))))),bNF_Wellorder_ordIso(set(A),fun(A,bool)))) ).

% card_of_Pow_Func
tff(fact_7583_type__copy__set__bd,axiom,
    ! [A: $tType,D: $tType,C: $tType,B: $tType,S: fun(A,set(B)),Bd: set(product_prod(C,C)),Rep: fun(D,A),X: D] :
      ( ! [Y3: A] : pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),S,Y3))),Bd)),bNF_Wellorder_ordLeq(B,C)))
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),aa(fun(D,A),fun(D,set(B)),comp(A,set(B),D,S),Rep),X))),Bd)),bNF_Wellorder_ordLeq(B,C))) ) ).

% type_copy_set_bd
tff(fact_7584_antisym__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( antisym(A,R3)
     => antisym(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% antisym_Restr
tff(fact_7585_card__of__ordLeq2,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => ( ? [G7: fun(B,A)] : aa(set(B),set(A),image2(B,A,G7),B5) = A6
      <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B))) ) ) ).

% card_of_ordLeq2
tff(fact_7586_surj__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,B5: set(A),F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F3),A6)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ) ).

% surj_imp_ordLeq
tff(fact_7587_card__of__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B2: B] :
      ( ( A6 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_singl_ordLeq
tff(fact_7588_card__of__ordLess2,axiom,
    ! [A: $tType,B: $tType,B5: set(A),A6: set(B)] :
      ( ( B5 != bot_bot(set(A)) )
     => ( ~ ? [F11: fun(B,A)] : aa(set(B),set(A),image2(B,A,F11),A6) = B5
      <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_We4044943003108391690rdLess(B,A))) ) ) ).

% card_of_ordLess2
tff(fact_7589_internalize__card__of__ordLeq2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),C4: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,C4))),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B11: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B11),C4))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B11))),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B11)),bNF_Ca6860139660246222851ard_of(B,C4))),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq2
tff(fact_7590_card__of__Field__ordLess,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R3))),R3)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_Field_ordLess
tff(fact_7591_card__of__Func__UNIV,axiom,
    ! [B: $tType,A: $tType,B5: set(B)] : pp(aa(set(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),bool,member(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),bNF_Ca6860139660246222851ard_of(fun(A,B),bNF_Wellorder_Func(A,B,top_top(set(A)),B5))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_ase(set(B),fun(fun(A,B),bool),B5))))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B)))) ).

% card_of_Func_UNIV
tff(fact_7592_ordLeq__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,field2(A,R3),aTP_Lamp_abr(set(C),fun(A,set(C)),C4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,field2(B,R6),aTP_Lamp_abq(set(C),fun(B,set(C)),C4))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% ordLeq_Times_mono1
tff(fact_7593_ordLeq__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),A6: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,A6,aTP_Lamp_asf(set(product_prod(A,A)),fun(C,set(A)),R3)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,A6,aTP_Lamp_asg(set(product_prod(B,B)),fun(C,set(B)),R6))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% ordLeq_Times_mono2
tff(fact_7594_card__of__ordLeq,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ? [F11: fun(A,B)] :
          ( inj_on(A,B,F11,A6)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F11),A6)),B5)) )
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B))) ) ).

% card_of_ordLeq
tff(fact_7595_internalize__card__of__ordLeq,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B11: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B11),field2(B,R3)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B11))),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B11)),R3)),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq
tff(fact_7596_card__of__ordLess,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ~ ? [F11: fun(A,B)] :
            ( inj_on(A,B,F11,A6)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F11),A6)),B5)) )
    <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_We4044943003108391690rdLess(B,A))) ) ).

% card_of_ordLess
tff(fact_7597_card__of__UNION__ordLeq__infinite,axiom,
    ! [B: $tType,A: $tType,C: $tType,B5: set(A),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),B5))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(B,A)))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),I5))
             => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(C,A))) )
         => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A6),I5)))),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(C,A))) ) ) ) ).

% card_of_UNION_ordLeq_infinite
tff(fact_7598_regularCard__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( bNF_Ca7133664381575040944arCard(A,R3)
    <=> ! [K9: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K9),field2(A,R3)))
            & bNF_Ca7293521722713021262ofinal(A,K9,R3) )
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,K9)),R3)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% regularCard_def
tff(fact_7599_Card__order__iff__Restr__underS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R3),R3)
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),field2(A,R3)))
           => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,order_underS(A,R3,X4),aa(A,fun(A,set(A)),aTP_Lamp_arw(set(product_prod(A,A)),fun(A,fun(A,set(A))),R3),X4)))),bNF_Ca6860139660246222851ard_of(A,field2(A,R3)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).

% Card_order_iff_Restr_underS
tff(fact_7600_card__of__card__order__on,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),bNF_Ca6860139660246222851ard_of(A,A6))) ).

% card_of_card_order_on
tff(fact_7601_card__of__Card__order,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Ca6860139660246222851ard_of(A,A6))) ).

% card_of_Card_order
tff(fact_7602_card__of__unique,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R3))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_unique
tff(fact_7603_Card__order__ordIso2,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
       => pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R6)),R6)) ) ) ).

% Card_order_ordIso2
tff(fact_7604_Card__order__ordIso,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordIso(B,A)))
       => pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R6)),R6)) ) ) ).

% Card_order_ordIso
tff(fact_7605_card__order__on__ordIso,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),R6: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R3))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R6))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R6)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% card_order_on_ordIso
tff(fact_7606_exists__minim__Card__order,axiom,
    ! [A: $tType,R2: set(set(product_prod(A,A)))] :
      ( ( R2 != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X3),R2))
           => pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,X3)),X3)) )
       => ? [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X3),R2))
            & ! [Xa: set(product_prod(A,A))] :
                ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),Xa),R2))
               => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% exists_minim_Card_order
tff(fact_7607_card__order__on__def,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R3))
    <=> ( order_well_order_on(A,A6,R3)
        & ! [R10: set(product_prod(A,A))] :
            ( order_well_order_on(A,A6,R10)
           => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),R10)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ).

% card_order_on_def
tff(fact_7608_card__order__on__well__order__on,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R3))
     => order_well_order_on(A,A6,R3) ) ).

% card_order_on_well_order_on
tff(fact_7609_Card__order__infinite__not__under,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
       => ~ ? [A14: A] : field2(A,R3) = aa(A,set(A),order_under(A,R3),A14) ) ) ).

% Card_order_infinite_not_under
tff(fact_7610_card__order__on__Card__order,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R3))
     => ( ( A6 = field2(A,R3) )
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3)) ) ) ).

% card_order_on_Card_order
tff(fact_7611_card__order__on,axiom,
    ! [A: $tType,A6: set(A)] :
    ? [X_1: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),X_1)) ).

% card_order_on
tff(fact_7612_Card__order__wo__rel,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => bNF_Wellorder_wo_rel(A,R3) ) ).

% Card_order_wo_rel
tff(fact_7613_Card__order__trans,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: A,Y: A,Z4: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ( X != Y )
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R3))
         => ( ( Y != Z4 )
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z4)),R3))
             => ( ( X != Z4 )
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z4)),R3)) ) ) ) ) ) ) ).

% Card_order_trans
tff(fact_7614_infinite__Card__order__limit,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
       => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),field2(A,R3)))
              & ( A3 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),R3)) ) ) ) ) ).

% infinite_Card_order_limit
tff(fact_7615_Card__order__Times2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ( A6 != bot_bot(set(B)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),R3),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A6,aTP_Lamp_ash(set(product_prod(A,A)),fun(B,set(A)),R3))))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ) ) ).

% Card_order_Times2
tff(fact_7616_Card__order__Times1,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),B5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ( B5 != bot_bot(set(B)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),R3),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R3),aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))),bNF_Wellorder_ordLeq(A,product_prod(A,B)))) ) ) ).

% Card_order_Times1
tff(fact_7617_Card__order__Times__same__infinite,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
       => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,field2(A,R3),aTP_Lamp_aiy(set(product_prod(A,A)),fun(A,set(A)),R3)))),R3)),bNF_Wellorder_ordLeq(product_prod(A,A),A))) ) ) ).

% Card_order_Times_same_infinite
tff(fact_7618_Card__order__Pow,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),bNF_Ca6860139660246222851ard_of(set(A),pow(A,field2(A,R3))))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% Card_order_Pow
tff(fact_7619_Card__order__iff__ordLeq__card__of,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Ca6860139660246222851ard_of(A,field2(A,R3)))),bNF_Wellorder_ordLeq(A,A))) ) ).

% Card_order_iff_ordLeq_card_of
tff(fact_7620_ordIso__card__of__imp__Card__order,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3)) ) ).

% ordIso_card_of_imp_Card_order
tff(fact_7621_Card__order__iff__ordIso__card__of,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),bNF_Ca6860139660246222851ard_of(A,field2(A,R3)))),bNF_Wellorder_ordIso(A,A))) ) ).

% Card_order_iff_ordIso_card_of
tff(fact_7622_card__of__Field__ordIso,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R3))),R3)),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_Field_ordIso
tff(fact_7623_Card__order__empty,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ).

% Card_order_empty
tff(fact_7624_card__of__ordIso__finite__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordIso(A,B)))
       => ( pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
        <=> pp(aa(set(B),bool,finite_finite2(B),A6)) ) ) ) ).

% card_of_ordIso_finite_Field
tff(fact_7625_card__of__underS,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R3)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,order_underS(A,R3,A3))),R3)),bNF_We4044943003108391690rdLess(A,A))) ) ) ).

% card_of_underS
tff(fact_7626_regularCard__UNION,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),As3: fun(A,set(B)),B5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( bNF_Ca7133664381575040944arCard(A,R3)
       => ( bNF_Ca3754400796208372196lChain(A,set(B),R3,As3)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As3),field2(A,R3)))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R3)),bNF_We4044943003108391690rdLess(B,A)))
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),field2(A,R3)))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(A,set(B),As3,X3))) ) ) ) ) ) ) ).

% regularCard_UNION
tff(fact_7627_Card__order__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),B2: B] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ( field2(A,R3) != bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B2),bot_bot(set(B))))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ).

% Card_order_singl_ordLeq
tff(fact_7628_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),A6: set(B),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R3)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R3)),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
           => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
tff(fact_7629_card__of__empty1,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A))] :
      ( ( order_well_order_on(A,field2(A,R3),R3)
        | pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R3)),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_empty1
tff(fact_7630_Card__order__Times__infinite,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => ( ( field2(B,P3) != bot_bot(set(B)) )
         => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R3)),bNF_Wellorder_ordLeq(B,A)))
           => ( pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R3),aTP_Lamp_asi(set(product_prod(B,B)),fun(A,set(B)),P3)))),R3)),bNF_Wellorder_ordIso(product_prod(A,B),A)))
              & pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,field2(B,P3),aTP_Lamp_ash(set(product_prod(A,A)),fun(B,set(A)),R3)))),R3)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ) ).

% Card_order_Times_infinite
tff(fact_7631_card__of__Times__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),A6: set(B),B5: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R3)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),R3)),bNF_Wellorder_ordLeq(C,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,A6,aTP_Lamp_abq(set(C),fun(B,set(C)),B5)))),R3)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Times_ordLeq_infinite_Field
tff(fact_7632_card__of__Sigma__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R3)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),R3)),bNF_Wellorder_ordLeq(C,A))) )
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A6))),R3)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Sigma_ordLeq_infinite_Field
tff(fact_7633_card__of__UNION__ordLeq__infinite__Field,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R3)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),R3)),bNF_Wellorder_ordLeq(C,A))) )
           => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A6),I5)))),R3)),bNF_Wellorder_ordLeq(C,A))) ) ) ) ) ).

% card_of_UNION_ordLeq_infinite_Field
tff(fact_7634_ex__toCard__pred,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3))
       => ? [X_1: fun(A,B)] : pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A6,R3),X_1)) ) ) ).

% ex_toCard_pred
tff(fact_7635_toCard__pred__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A6,R3),F3))
    <=> ( inj_on(A,B,F3,A6)
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A6)),field2(B,R3)))
        & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)) ) ) ).

% toCard_pred_def
tff(fact_7636_toCard__pred__toCard,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3))
       => pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A6,R3),bNF_Greatest_toCard(A,B,A6,R3))) ) ) ).

% toCard_pred_toCard
tff(fact_7637_cardSuc__UNION,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),As3: fun(set(A),set(B)),B5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
       => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R3),As3)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As3),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R3))))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R3)),bNF_Wellorder_ordLeq(B,A)))
             => ? [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X3),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R3))))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),As3,X3))) ) ) ) ) ) ) ).

% cardSuc_UNION
tff(fact_7638_cardSuc__Card__order,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R3))),bNF_Ca8387033319878233205ardSuc(A,R3))) ) ).

% cardSuc_Card_order
tff(fact_7639_infinite__cardSuc__regularCard,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => bNF_Ca7133664381575040944arCard(set(A),bNF_Ca8387033319878233205ardSuc(A,R3)) ) ) ).

% infinite_cardSuc_regularCard
tff(fact_7640_card__of__cardSuc__finite,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,bNF_Ca6860139660246222851ard_of(A,A6)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),A6)) ) ).

% card_of_cardSuc_finite
tff(fact_7641_cardSuc__ordLeq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),bNF_Ca8387033319878233205ardSuc(A,R3))),bNF_Wellorder_ordLeq(A,set(A)))) ) ).

% cardSuc_ordLeq
tff(fact_7642_cardSuc__finite,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R3))))
      <=> pp(aa(set(A),bool,finite_finite2(A),field2(A,R3))) ) ) ).

% cardSuc_finite
tff(fact_7643_cardSuc__greater,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),bNF_Ca8387033319878233205ardSuc(A,R3))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% cardSuc_greater
tff(fact_7644_cardSuc__least,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R6)),R6))
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
         => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R3)),R6)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_least
tff(fact_7645_cardSuc__ordLess__ordLeq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R6)),R6))
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_We4044943003108391690rdLess(A,B)))
        <=> pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R3)),R6)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_ordLess_ordLeq
tff(fact_7646_cardSuc__mono__ordLeq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R6)),R6))
       => ( pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R3)),bNF_Ca8387033319878233205ardSuc(B,R6))),bNF_Wellorder_ordLeq(set(A),set(B))))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% cardSuc_mono_ordLeq
tff(fact_7647_cardSuc__invar__ordIso,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R6)),R6))
       => ( pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R3)),bNF_Ca8387033319878233205ardSuc(B,R6))),bNF_Wellorder_ordIso(set(A),set(B))))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B))) ) ) ) ).

% cardSuc_invar_ordIso
tff(fact_7648_cardSuc__least__aux,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(set(A),set(A)))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R6)),R6))
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),R6)),bNF_We4044943003108391690rdLess(A,set(A))))
         => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),bNF_Ca8387033319878233205ardSuc(A,R3)),R6)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% cardSuc_least_aux
tff(fact_7649_cardSuc__ordLeq__ordLess,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R6)),R6))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(B,B)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),R6),bNF_Ca8387033319878233205ardSuc(A,R3))),bNF_We4044943003108391690rdLess(B,set(A))))
        <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R6),R3)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% cardSuc_ordLeq_ordLess
tff(fact_7650_toCard__inj,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(B,B)),X: A,Y: A] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3))
       => ( pp(aa(set(A),bool,member(A,X),A6))
         => ( pp(aa(set(A),bool,member(A,Y),A6))
           => ( ( aa(A,B,bNF_Greatest_toCard(A,B,A6,R3),X) = aa(A,B,bNF_Greatest_toCard(A,B,A6,R3),Y) )
            <=> ( X = Y ) ) ) ) ) ) ).

% toCard_inj
tff(fact_7651_fromCard__toCard,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(B,B)),B2: A] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3))
       => ( pp(aa(set(A),bool,member(A,B2),A6))
         => ( bNF_Gr5436034075474128252omCard(A,B,A6,R3,aa(A,B,bNF_Greatest_toCard(A,B,A6,R3),B2)) = B2 ) ) ) ) ).

% fromCard_toCard
tff(fact_7652_isCardSuc__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(set(A),set(A)))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca6246979054910435723ardSuc(A,R3),R6))
    <=> ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R6)),R6))
        & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),R6)),bNF_We4044943003108391690rdLess(A,set(A))))
        & ! [R11: set(product_prod(set(A),set(A)))] :
            ( ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R11)),R11))
              & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R3),R11)),bNF_We4044943003108391690rdLess(A,set(A)))) )
           => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),R6),R11)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% isCardSuc_def
tff(fact_7653_exists__isCardSuc,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ? [X_1: set(product_prod(set(A),set(A)))] : pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca6246979054910435723ardSuc(A,R3),X_1)) ) ).

% exists_isCardSuc
tff(fact_7654_cardSuc__isCardSuc,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca6246979054910435723ardSuc(A,R3),bNF_Ca8387033319878233205ardSuc(A,R3))) ) ).

% cardSuc_isCardSuc
tff(fact_7655_comp__single__set__bd,axiom,
    ! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set(product_prod(A,A)),Fset: fun(B,set(C)),Gset: fun(D,set(B)),Gbd: set(product_prod(E,E)),X: D] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,Fbd)),Fbd))
     => ( ! [X3: B] : pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Fset,X3))),Fbd)),bNF_Wellorder_ordLeq(C,A)))
       => ( ! [X3: D] : pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),Gset,X3))),Gbd)),bNF_Wellorder_ordLeq(B,E)))
         => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(C,C)),fun(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),product_Pair(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Fset),aa(D,set(B),Gset,X))))),bNF_Cardinal_cprod(E,A,Gbd,Fbd))),bNF_Wellorder_ordLeq(C,product_prod(E,A)))) ) ) ) ).

% comp_single_set_bd
tff(fact_7656_cardSuc__UNION__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),As3: fun(set(A),set(B)),B5: set(B)] :
      ( ( bNF_Ca4139267488887388095finite(A,R3)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3)) )
     => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R3),As3)
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As3),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R3))))))
         => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R3)),bNF_Wellorder_ordLeq(B,A)))
           => ? [X3: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),X3),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R3))))
                & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),As3,X3))) ) ) ) ) ) ).

% cardSuc_UNION_Cinfinite
tff(fact_7657_cprod__def,axiom,
    ! [A: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B))] : bNF_Cardinal_cprod(A,B,R12,R23) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R12),aTP_Lamp_asi(set(product_prod(B,B)),fun(A,set(B)),R23))) ).

% cprod_def
tff(fact_7658_Cinfinite__limit2,axiom,
    ! [A: $tType,X1: A,R3: set(product_prod(A,A)),X2: A] :
      ( pp(aa(set(A),bool,member(A,X1),field2(A,R3)))
     => ( pp(aa(set(A),bool,member(A,X2),field2(A,R3)))
       => ( ( bNF_Ca4139267488887388095finite(A,R3)
            & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3)) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),field2(A,R3)))
              & ( X1 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X3)),R3))
              & ( X2 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X3)),R3)) ) ) ) ) ).

% Cinfinite_limit2
tff(fact_7659_Cinfinite__limit,axiom,
    ! [A: $tType,X: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,X),field2(A,R3)))
     => ( ( bNF_Ca4139267488887388095finite(A,R3)
          & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3)) )
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),field2(A,R3)))
            & ( X != X3 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X3)),R3)) ) ) ) ).

% Cinfinite_limit
tff(fact_7660_cprod__dup,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(B,B)),P9: set(product_prod(C,C))] :
      ( bNF_Ca4139267488887388095finite(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => ( pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Cardinal_cprod(B,C,P3,P9)),bNF_Cardinal_cprod(A,A,R3,R3))),bNF_Wellorder_ordIso(product_prod(B,C),product_prod(A,A))))
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Cardinal_cprod(B,C,P3,P9)),R3)),bNF_Wellorder_ordIso(product_prod(B,C),A))) ) ) ) ).

% cprod_dup
tff(fact_7661_Un__Cinfinite__bound,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(B,B)),B5: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B5)),R3)),bNF_Wellorder_ordLeq(A,B)))
       => ( ( bNF_Ca4139267488887388095finite(B,R3)
            & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)) )
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))),R3)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% Un_Cinfinite_bound
tff(fact_7662_Cinfinite__limit__finite,axiom,
    ! [A: $tType,X6: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),field2(A,R3)))
       => ( ( bNF_Ca4139267488887388095finite(A,R3)
            & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3)) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),field2(A,R3)))
              & ! [Xa: A] :
                  ( pp(aa(set(A),bool,member(A,Xa),X6))
                 => ( ( Xa != X3 )
                    & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X3)),R3)) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
tff(fact_7663_UNION__Cinfinite__bound,axiom,
    ! [A: $tType,B: $tType,C: $tType,I5: set(A),R3: set(product_prod(B,B)),A6: fun(A,set(C))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,I5)),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),I5))
           => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),A6,X3))),R3)),bNF_Wellorder_ordLeq(C,B))) )
       => ( ( bNF_Ca4139267488887388095finite(B,R3)
            & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)) )
         => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A6),I5)))),R3)),bNF_Wellorder_ordLeq(C,B))) ) ) ) ).

% UNION_Cinfinite_bound
tff(fact_7664_card__of__Csum__Times_H,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),I5: set(B),A6: fun(B,set(C))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,member(B,X3),I5))
           => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),R3)),bNF_Wellorder_ordLeq(C,A))) )
       => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_Csum(B,C,bNF_Ca6860139660246222851ard_of(B,I5),aTP_Lamp_asj(fun(B,set(C)),fun(B,set(product_prod(C,C))),A6))),bNF_Cardinal_cprod(B,A,bNF_Ca6860139660246222851ard_of(B,I5),R3))),bNF_Wellorder_ordLeq(product_prod(B,C),product_prod(B,A)))) ) ) ).

% card_of_Csum_Times'
tff(fact_7665_card__of__Csum__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B5: set(C)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),I5))
         => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A6,X3))),bNF_Ca6860139660246222851ard_of(C,B5))),bNF_Wellorder_ordLeq(B,C))) )
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I5),aTP_Lamp_ask(fun(A,set(B)),fun(A,set(product_prod(B,B))),A6))),bNF_Cardinal_cprod(A,C,bNF_Ca6860139660246222851ard_of(A,I5),bNF_Ca6860139660246222851ard_of(C,B5)))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C)))) ) ).

% card_of_Csum_Times
tff(fact_7666_Csum__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),Rs: fun(A,set(product_prod(B,B)))] : bNF_Cardinal_Csum(A,B,R3,Rs) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R3),aTP_Lamp_asl(fun(A,set(product_prod(B,B))),fun(A,set(B)),Rs))) ).

% Csum_def
tff(fact_7667_SIGMA__CSUM,axiom,
    ! [B: $tType,A: $tType,I5: set(A),As3: fun(A,set(B))] : bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I5,As3)) = bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I5),aTP_Lamp_ask(fun(A,set(B)),fun(A,set(product_prod(B,B))),As3)) ).

% SIGMA_CSUM
tff(fact_7668_card__of__Plus__Times__aux,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,A6: set(A),B5: set(B)] :
      ( ( ( A1 != A22 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A22),bot_bot(set(A))))),A6)) )
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B5))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).

% card_of_Plus_Times_aux
tff(fact_7669_infinite__iff__natLeq__ordLeq,axiom,
    ! [A: $tType,A6: set(A)] :
      ~ ( pp(aa(set(A),bool,finite_finite2(A),A6))
      <=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca8665028551170535155natLeq),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(nat,A))) ) ).

% infinite_iff_natLeq_ordLeq
tff(fact_7670_ordIso__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),P3: set(product_prod(C,C)),P9: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P9)),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R3),field2(C,P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R6),field2(D,P9))))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordIso_Plus_cong
tff(fact_7671_ordIso__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R3),C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R6),C4)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordIso_Plus_cong1
tff(fact_7672_ordIso__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),A6: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A6,field2(A,R3)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A6,field2(B,R6))))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordIso_Plus_cong2
tff(fact_7673_ordLeq__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),P3: set(product_prod(C,C)),P9: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P9)),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R3),field2(C,P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R6),field2(D,P9))))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordLeq_Plus_mono
tff(fact_7674_ordLeq__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R3),C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R6),C4)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordLeq_Plus_mono1
tff(fact_7675_ordLeq__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(B,B)),A6: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R6)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A6,field2(A,R3)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A6,field2(B,R6))))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordLeq_Plus_mono2
tff(fact_7676_natLeq__Card__order,axiom,
    pp(aa(set(product_prod(nat,nat)),bool,bNF_Ca8970107618336181345der_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq)) ).

% natLeq_Card_order
tff(fact_7677_Card__order__Plus2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),R3),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A6,field2(A,R3))))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ) ).

% Card_order_Plus2
tff(fact_7678_Card__order__Plus1,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,A)),B5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),R3),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R3),B5)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ) ).

% Card_order_Plus1
tff(fact_7679_card__of__Un__Plus__ordLeq,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A6,B5)))),bNF_Wellorder_ordLeq(A,sum_sum(A,A)))) ).

% card_of_Un_Plus_ordLeq
tff(fact_7680_card__of__Times__Plus__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),bool,member(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),fun(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),product_Pair(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),bNF_Ca6860139660246222851ard_of(product_prod(A,sum_sum(B,C)),product_Sigma(A,sum_sum(B,C),A6,aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_asm(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),B5),C4)))),bNF_Ca6860139660246222851ard_of(sum_sum(product_prod(A,B),product_prod(A,C)),sum_Plus(product_prod(A,B),product_prod(A,C),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5)),product_Sigma(A,C,A6,aTP_Lamp_abr(set(C),fun(A,set(C)),C4)))))),bNF_Wellorder_ordIso(product_prod(A,sum_sum(B,C)),sum_sum(product_prod(A,B),product_prod(A,C))))) ).

% card_of_Times_Plus_distrib
tff(fact_7681_card__of__Plus__Times__bool,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),bool,member(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),aa(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),fun(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),product_Pair(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A6,A6))),bNF_Ca6860139660246222851ard_of(product_prod(A,bool),product_Sigma(A,bool,A6,aTP_Lamp_asn(A,set(bool)))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,bool)))) ).

% card_of_Plus_Times_bool
tff(fact_7682_card__of__Plus__commute,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] : pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B5,A6)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A)))) ).

% card_of_Plus_commute
tff(fact_7683_card__of__Plus__assoc,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C)] : pp(aa(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),bool,member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),fun(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),bNF_Ca6860139660246222851ard_of(sum_sum(sum_sum(A,B),C),sum_Plus(sum_sum(A,B),C,sum_Plus(A,B,A6,B5),C4))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,C)),sum_Plus(A,sum_sum(B,C),A6,sum_Plus(B,C,B5,C4))))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C))))) ).

% card_of_Plus_assoc
tff(fact_7684_card__of__Plus2,axiom,
    ! [B: $tType,A: $tType,B5: set(A),A6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A6,B5)))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ).

% card_of_Plus2
tff(fact_7685_card__of__Plus1,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B5)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ).

% card_of_Plus1
tff(fact_7686_card__of__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C),D5: set(D)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C4)),bNF_Ca6860139660246222851ard_of(D,D5))),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B5,D5)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_mono
tff(fact_7687_card__of__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B5,C4)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_mono1
tff(fact_7688_card__of__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C4,A6))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C4,B5)))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_mono2
tff(fact_7689_card__of__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C),D5: set(D)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C4)),bNF_Ca6860139660246222851ard_of(D,D5))),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B5,D5)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_cong
tff(fact_7690_card__of__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C4))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B5,C4)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_cong1
tff(fact_7691_card__of__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A6: set(A),B5: set(B),C4: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C4,A6))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C4,B5)))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_cong2
tff(fact_7692_card__of__Plus__empty2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A6)))),bNF_Wellorder_ordIso(A,sum_sum(B,A)))) ).

% card_of_Plus_empty2
tff(fact_7693_card__of__Plus__empty1,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,bot_bot(set(B)))))),bNF_Wellorder_ordIso(A,sum_sum(A,B)))) ).

% card_of_Plus_empty1
tff(fact_7694_natLeq__antisym,axiom,
    antisym(nat,bNF_Ca8665028551170535155natLeq) ).

% natLeq_antisym
tff(fact_7695_card__Plus,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B5))
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A6,B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B5)) ) ) ) ).

% card_Plus
tff(fact_7696_natLeq__natLess__Id,axiom,
    bNF_Ca8459412986667044542atLess = aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),minus_minus(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),id2(nat)) ).

% natLeq_natLess_Id
tff(fact_7697_natLeq__trans,axiom,
    trans(nat,bNF_Ca8665028551170535155natLeq) ).

% natLeq_trans
tff(fact_7698_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N)) ).

% natLeq_underS_less
tff(fact_7699_natLeq__Total,axiom,
    total_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Total
tff(fact_7700_natLeq__Linear__order,axiom,
    order_679001287576687338der_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Linear_order
tff(fact_7701_natLeq__Preorder,axiom,
    order_preorder_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Preorder
tff(fact_7702_natLeq__Refl,axiom,
    refl_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Refl
tff(fact_7703_Field__natLeq,axiom,
    field2(nat,bNF_Ca8665028551170535155natLeq) = top_top(set(nat)) ).

% Field_natLeq
tff(fact_7704_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less_eq(nat))) ).

% natLeq_def
tff(fact_7705_card__Plus__conv__if,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ( ( pp(aa(set(A),bool,finite_finite2(A),A6))
          & pp(aa(set(B),bool,finite_finite2(B),B5)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A6,B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B5)) ) )
      & ( ~ ( pp(aa(set(A),bool,finite_finite2(A),A6))
            & pp(aa(set(B),bool,finite_finite2(B),B5)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A6,B5)) = zero_zero(nat) ) ) ) ).

% card_Plus_conv_if
tff(fact_7706_card__of__Plus__infinite2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B5,A6))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ).

% card_of_Plus_infinite2
tff(fact_7707_card__of__Plus__infinite1,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B5))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ).

% card_of_Plus_infinite1
tff(fact_7708_card__of__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B5))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
          & pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B5,A6))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ).

% card_of_Plus_infinite
tff(fact_7709_card__of__Plus__ordLess__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,C4: set(A),A6: set(B),B5: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),C4))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),bNF_Ca6860139660246222851ard_of(A,C4))),bNF_We4044943003108391690rdLess(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),bNF_Ca6860139660246222851ard_of(A,C4))),bNF_We4044943003108391690rdLess(C,A)))
         => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A6,B5))),bNF_Ca6860139660246222851ard_of(A,C4))),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ).

% card_of_Plus_ordLess_infinite
tff(fact_7710_natLeq__Well__order,axiom,
    order_well_order_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Well_order
tff(fact_7711_finite__iff__ordLess__natLeq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(A,A)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(A,A)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca8665028551170535155natLeq)),bNF_We4044943003108391690rdLess(A,nat))) ) ).

% finite_iff_ordLess_natLeq
tff(fact_7712_Restr__natLeq2,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,N),aTP_Lamp_aso(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq2
tff(fact_7713_Restr__natLeq,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),N)),aTP_Lamp_asp(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq
tff(fact_7714_Card__order__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R3)),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R3),field2(B,P3)))),R3)),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
            & pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,field2(B,P3),field2(A,R3)))),R3)),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ) ).

% Card_order_Plus_infinite
tff(fact_7715_card__of__nat,axiom,
    pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca8665028551170535155natLeq)),bNF_Wellorder_ordIso(nat,nat))) ).

% card_of_nat
tff(fact_7716_card__of__Plus__Times,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,A6: set(A),B1: B,B22: B,B5: set(B)] :
      ( ( ( A1 != A22 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A22),bot_bot(set(A))))),A6)) )
     => ( ( ( B1 != B22 )
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B1),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B22),bot_bot(set(B))))),B5)) )
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B5))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).

% card_of_Plus_Times
tff(fact_7717_card__of__Field__natLeq,axiom,
    pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,field2(nat,bNF_Ca8665028551170535155natLeq))),bNF_Ca8665028551170535155natLeq)),bNF_Wellorder_ordIso(nat,nat))) ).

% card_of_Field_natLeq
tff(fact_7718_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),A6: set(B),B5: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R3)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),R3)),bNF_Wellorder_ordLeq(C,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
           => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A6,B5))),R3)),bNF_Wellorder_ordLeq(sum_sum(B,C),A))) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
tff(fact_7719_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),A6: set(B),B5: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R3)),bNF_We4044943003108391690rdLess(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),R3)),bNF_We4044943003108391690rdLess(C,A)))
           => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A6,B5))),R3)),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
tff(fact_7720_ord__to__filter__compat,axiom,
    ! [A: $tType,R0: set(product_prod(A,A))] : bNF_Wellorder_compat(set(product_prod(A,A)),set(A),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),inf_inf(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),bNF_We4044943003108391690rdLess(A,A)),product_Sigma(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A))),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),set(set(product_prod(A,A)))),insert(set(product_prod(A,A))),R0),bot_bot(set(set(product_prod(A,A)))))),aTP_Lamp_asq(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),R0))),bNF_We413866401316099525erIncl(A,R0),bNF_We8469521843155493636filter(A,R0)) ).

% ord_to_filter_compat
tff(fact_7721_cINF__UNION,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A6: set(C),B5: fun(C,set(D)),F3: fun(D,B)] :
          ( ( A6 != bot_bot(set(C)) )
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,member(C,X3),A6))
               => ( aa(C,set(D),B5,X3) != bot_bot(set(D)) ) )
           => ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_arn(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B5),F3)),A6)))
             => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,F3),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B5),A6)))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_asr(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B5),F3)),A6)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_7722_bdd__below__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A6: set(A)] : condit1013018076250108175_below(A,A6) ) ).

% bdd_below_bot
tff(fact_7723_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),M6: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M6),X3)) )
         => condit1013018076250108175_below(A,A6) ) ) ).

% bdd_below.I
tff(fact_7724_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),M: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X3)) )
         => condit1013018076250108175_below(A,A6) ) ) ).

% bdd_belowI
tff(fact_7725_bdd__below__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit1013018076250108175_below(A,bot_bot(set(A))) ) ).

% bdd_below_empty
tff(fact_7726_bdd__below__insert,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: A,A6: set(A)] :
          ( condit1013018076250108175_below(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),A6))
        <=> condit1013018076250108175_below(A,A6) ) ) ).

% bdd_below_insert
tff(fact_7727_bdd__below__Un,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( condit1013018076250108175_below(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))
        <=> ( condit1013018076250108175_below(A,A6)
            & condit1013018076250108175_below(A,B5) ) ) ) ).

% bdd_below_Un
tff(fact_7728_bdd__below__Icc,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit1013018076250108175_below(A,set_or1337092689740270186AtMost(A,A3,B2)) ) ).

% bdd_below_Icc
tff(fact_7729_bdd__below__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit1013018076250108175_below(A,set_or7035219750837199246ssThan(A,A3,B2)) ) ).

% bdd_below_Ico
tff(fact_7730_bdd__below__Ioc,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit1013018076250108175_below(A,set_or3652927894154168847AtMost(A,A3,B2)) ) ).

% bdd_below_Ioc
tff(fact_7731_bdd__below__Ioo,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] : condit1013018076250108175_below(A,set_or5935395276787703475ssThan(A,A3,B2)) ) ).

% bdd_below_Ioo
tff(fact_7732_bdd__below__Ici,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : condit1013018076250108175_below(A,aa(A,set(A),set_ord_atLeast(A),A3)) ) ).

% bdd_below_Ici
tff(fact_7733_bdd__below__Ioi,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : condit1013018076250108175_below(A,aa(A,set(A),set_ord_greaterThan(A),A3)) ) ).

% bdd_below_Ioi
tff(fact_7734_bdd__below__image__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ass(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A6))
        <=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
            & condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G3),A6)) ) ) ) ).

% bdd_below_image_inf
tff(fact_7735_bdd__below__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: set(A)] :
          ( condit1013018076250108175_below(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),X6))
        <=> condit941137186595557371_above(A,X6) ) ) ).

% bdd_below_uminus
tff(fact_7736_bdd__above__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X6: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),image2(A,A,uminus_uminus(A)),X6))
        <=> condit1013018076250108175_below(A,X6) ) ) ).

% bdd_above_uminus
tff(fact_7737_bdd__below__UN,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [I5: set(B),A6: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( condit1013018076250108175_below(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A6),I5)))
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),I5))
               => condit1013018076250108175_below(A,aa(B,set(A),A6,X4)) ) ) ) ) ).

% bdd_below_UN
tff(fact_7738_compat__def,axiom,
    ! [A: $tType,A2: $tType,R3: set(product_prod(A,A)),R6: set(product_prod(A2,A2)),F3: fun(A,A2)] :
      ( bNF_Wellorder_compat(A,A2,R3,R6,F3)
    <=> ! [A9: A,B7: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),B7)),R3))
         => pp(aa(set(product_prod(A2,A2)),bool,member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,F3,A9)),aa(A,A2,F3,B7))),R6)) ) ) ).

% compat_def
tff(fact_7739_bdd__below__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => condit1013018076250108175_below(A,A6) ) ) ).

% bdd_below_finite
tff(fact_7740_cInf__lower,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A)] :
          ( pp(aa(set(A),bool,member(A,X),X6))
         => ( condit1013018076250108175_below(A,X6)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_lower
tff(fact_7741_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A),Y: A] :
          ( pp(aa(set(A),bool,member(A,X),X6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( condit1013018076250108175_below(A,X6)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y)) ) ) ) ) ).

% cInf_lower2
tff(fact_7742_bdd__below__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: set(A),A6: set(A)] :
          ( condit1013018076250108175_below(A,B5)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
           => condit1013018076250108175_below(A,A6) ) ) ) ).

% bdd_below_mono
tff(fact_7743_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( condit1013018076250108175_below(A,A6)
        <=> ? [M13: A] :
            ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M13),X4)) ) ) ) ).

% bdd_below.unfold
tff(fact_7744_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( condit1013018076250108175_below(A,A6)
         => ~ ! [M11: A] :
                ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M11),X5)) ) ) ) ).

% bdd_below.E
tff(fact_7745_bdd__below__Int2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: set(A),A6: set(A)] :
          ( condit1013018076250108175_below(A,B5)
         => condit1013018076250108175_below(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ) ) ).

% bdd_below_Int2
tff(fact_7746_bdd__below__Int1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),B5: set(A)] :
          ( condit1013018076250108175_below(A,A6)
         => condit1013018076250108175_below(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)) ) ) ).

% bdd_below_Int1
tff(fact_7747_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),X: B,U: A] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
         => ( pp(aa(set(B),bool,member(B,X),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),U)) ) ) ) ) ).

% cINF_lower2
tff(fact_7748_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),X: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
         => ( pp(aa(set(B),bool,member(B,X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(B,A,F3,X))) ) ) ) ).

% cINF_lower
tff(fact_7749_bdd__below__image__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( condit1013018076250108175_below(A,A6)
           => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A6)) ) ) ) ).

% bdd_below_image_mono
tff(fact_7750_bdd__belowI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A6: set(B),M: A,F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F3,X3))) )
         => condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6)) ) ) ).

% bdd_belowI2
tff(fact_7751_bdd__below_OI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A6: set(B),M6: A,F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M6),aa(B,A,F3,X3))) )
         => condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6)) ) ) ).

% bdd_below.I2
tff(fact_7752_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B5: set(A),A6: set(A)] :
          ( ( B5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A6)
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,member(A,B4),B5))
                 => ? [X5: A] :
                      ( pp(aa(set(A),bool,member(A,X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B4)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B5))) ) ) ) ) ).

% cInf_mono
tff(fact_7753_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),S))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X4)) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_7754_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Y: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X6)
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),X6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y)) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_7755_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),Y: A,I2: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))))
           => ( pp(aa(set(B),bool,member(B,I2),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F3,I2))) ) ) ) ) ).

% less_cINF_D
tff(fact_7756_bdd__below__image__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( order_antimono(A,B,F3)
         => ( condit941137186595557371_above(A,A6)
           => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A6)) ) ) ) ).

% bdd_below_image_antimono
tff(fact_7757_bdd__above__image__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( order_antimono(A,B,F3)
         => ( condit1013018076250108175_below(A,A6)
           => condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A6)) ) ) ) ).

% bdd_above_image_antimono
tff(fact_7758_le__cINF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))))
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_7759_cINF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B5: set(B),F3: fun(C,A),A6: set(C),G3: fun(B,A)] :
          ( ( B5 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,F3),A6))
           => ( ! [M5: B] :
                  ( pp(aa(set(B),bool,member(B,M5),B5))
                 => ? [X5: C] :
                      ( pp(aa(set(C),bool,member(C,X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X5)),aa(B,A,G3,M5))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),B5)))) ) ) ) ) ).

% cINF_mono
tff(fact_7760_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B5)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B5)),aa(set(A),A,complete_Inf_Inf(A),A6))) ) ) ) ) ).

% cInf_superset_mono
tff(fact_7761_cInf__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X6)
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ).

% cInf_insert
tff(fact_7762_cInf__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( condit1013018076250108175_below(A,X6)
         => ( ( ( X6 = bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)) = A3 ) )
            & ( ( X6 != bot_bot(set(A)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A3),X6)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ) ).

% cInf_insert_If
tff(fact_7763_cInf__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A6)
           => ( ( B5 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,B5)
               => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_7764_cINF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A6: set(B),F3: fun(B,A),A3: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),A3))
            <=> ? [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),A3)) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_7765_cINF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G3),A6))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),A6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ast(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A6)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_7766_cSUP__eq__cINF__D,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F3: fun(C,B),A6: set(C),A3: C] :
          ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F3),A6)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F3),A6)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,F3),A6))
           => ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F3),A6))
             => ( pp(aa(set(C),bool,member(C,A3),A6))
               => ( aa(C,B,F3,A3) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F3),A6)) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_7767_cINF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),G3: fun(B,A),B5: set(B),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,G3),B5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B5))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,member(B,X3),B5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,G3,X3)),aa(B,A,F3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G3),B5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6)))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_7768_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B5: set(A)] :
          ( condit1013018076250108175_below(A,A6)
         => ( condit1013018076250108175_below(A,B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5) != bot_bot(set(A)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B5)))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_7769_cINF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),A3: B] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A3),A6))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F3,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))) ) ) ) ) ).

% cINF_insert
tff(fact_7770_cINF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),B5: set(B)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A6))
           => ( ( B5 != bot_bot(set(B)) )
             => ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),B5))
               => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B5))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),B5))) ) ) ) ) ) ) ).

% cINF_union
tff(fact_7771_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A6)
           => ( condit1013018076250108175_below(A,A6)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ) ).

% cInf_le_cSup
tff(fact_7772_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( aa(set(A),A,complete_Inf_Inf(A),S) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_asu(set(A),fun(A,bool),S))) ) ) ) ) ).

% cInf_cSup
tff(fact_7773_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( condit1013018076250108175_below(A,A6)
           => ( ( A6 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A6))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A6)))) ) ) ) ) ).

% mono_cInf
tff(fact_7774_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,A6),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A6),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ars(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6)),I5)))) ) ) ) ) ).

% mono_cINF
tff(fact_7775_csum__dup,axiom,
    ! [A: $tType,C: $tType,B: $tType,R3: set(product_prod(A,A)),P3: set(product_prod(B,B)),P9: set(product_prod(C,C))] :
      ( bNF_Ca4139267488887388095finite(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R3)),R3))
       => ( pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),bool,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Cardinal_csum(B,C,P3,P9)),bNF_Cardinal_csum(A,A,R3,R3))),bNF_Wellorder_ordIso(sum_sum(B,C),sum_sum(A,A))))
         => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Cardinal_csum(B,C,P3,P9)),R3)),bNF_Wellorder_ordIso(sum_sum(B,C),A))) ) ) ) ).

% csum_dup
tff(fact_7776_ord__to__filter__def,axiom,
    ! [A: $tType,R0: set(product_prod(A,A)),R3: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),bNF_We8469521843155493636filter(A,R0),R3) = aa(set(A),set(A),image2(A,A,fChoice(fun(A,A),bNF_Wellorder_embed(A,A,R3,R0))),field2(A,R3)) ).

% ord_to_filter_def
tff(fact_7777_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,bool),fequal(A),X)) = X ).

% some_sym_eq_trivial
tff(fact_7778_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_ap(A,fun(A,bool),X)) = X ).

% some_eq_trivial
tff(fact_7779_some__equality,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P2,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => ( X3 = A3 ) )
       => ( fChoice(A,P2) = A3 ) ) ) ).

% some_equality
tff(fact_7780_some__insert__self,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S))),S) = S ) ) ).

% some_insert_self
tff(fact_7781_some__elem,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S))),S)) ) ).

% some_elem
tff(fact_7782_some__in__eq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6))),A6))
    <=> ( A6 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_7783_some__theI,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] :
      ( ? [A14: A,X_13: B] : pp(aa(B,bool,aa(A,fun(B,bool),P2,A14),X_13))
     => ( ! [B15: B,B24: B] :
            ( ? [A5: A] : pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B15))
           => ( ? [A5: A] : pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B24))
             => ( B15 = B24 ) ) )
       => pp(aa(B,bool,aa(A,fun(B,bool),P2,fChoice(A,aTP_Lamp_adq(fun(A,fun(B,bool)),fun(A,bool),P2))),the(B,aTP_Lamp_asv(fun(A,fun(B,bool)),fun(B,bool),P2)))) ) ) ).

% some_theI
tff(fact_7784_univ__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X6: set(B)] : bNF_Greatest_univ(B,A,F3,X6) = aa(B,A,F3,fChoice(B,aTP_Lamp_be(set(B),fun(B,bool),X6))) ).

% univ_def
tff(fact_7785_inv__on__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),X: B] : aa(B,A,inv_on(A,B,F3,A6),X) = fChoice(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ahw(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F3),A6),X)) ).

% inv_on_def
tff(fact_7786_some1__equality,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P2,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P2,Y3))
             => ( Y3 = X5 ) ) )
     => ( pp(aa(A,bool,P2,A3))
       => ( fChoice(A,P2) = A3 ) ) ) ).

% some1_equality
tff(fact_7787_some__eq__ex,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( pp(aa(A,bool,P2,fChoice(A,P2)))
    <=> ? [X_12: A] : pp(aa(A,bool,P2,X_12)) ) ).

% some_eq_ex
tff(fact_7788_someI2__bex,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool),Q2: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,member(A,X5),A6))
          & pp(aa(A,bool,P2,X5)) )
     => ( ! [X3: A] :
            ( ( pp(aa(set(A),bool,member(A,X3),A6))
              & pp(aa(A,bool,P2,X3)) )
           => pp(aa(A,bool,Q2,X3)) )
       => pp(aa(A,bool,Q2,fChoice(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2)))) ) ) ).

% someI2_bex
tff(fact_7789_someI2__ex,axiom,
    ! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
      ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => pp(aa(A,bool,Q2,X3)) )
       => pp(aa(A,bool,Q2,fChoice(A,P2))) ) ) ).

% someI2_ex
tff(fact_7790_someI__ex,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
     => pp(aa(A,bool,P2,fChoice(A,P2))) ) ).

% someI_ex
tff(fact_7791_someI2,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A,Q2: fun(A,bool)] :
      ( pp(aa(A,bool,P2,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => pp(aa(A,bool,Q2,X3)) )
       => pp(aa(A,bool,Q2,fChoice(A,P2))) ) ) ).

% someI2
tff(fact_7792_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P2: fun(A,bool),P4: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2)) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
          <=> pp(aa(A,bool,P4,X3)) )
       => ( ! [X_12: A] : pp(aa(A,bool,P4,X_12))
        <=> pp(aa(A,bool,P2,X)) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_7793_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P2: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2)) )
     => ( ! [X_12: A] : pp(aa(A,bool,P2,X_12))
      <=> pp(aa(A,bool,P2,X)) ) ) ).

% verit_sko_forall_indirect
tff(fact_7794_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P2: fun(A,bool),P4: fun(A,bool)] :
      ( ( X = fChoice(A,P2) )
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
          <=> pp(aa(A,bool,P4,X3)) )
       => ( ? [X_12: A] : pp(aa(A,bool,P4,X_12))
        <=> pp(aa(A,bool,P2,X)) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_7795_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P2: fun(A,bool)] :
      ( ( X = fChoice(A,P2) )
     => ( ? [X_12: A] : pp(aa(A,bool,P2,X_12))
      <=> pp(aa(A,bool,P2,X)) ) ) ).

% verit_sko_ex_indirect
tff(fact_7796_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B5: A,A6: A,P2: fun(A,bool)] :
      ( ( B5 = A6 )
     => ( ( fChoice(A,P2) = A6 )
      <=> ( fChoice(A,P2) = B5 ) ) ) ).

% verit_sko_forall''
tff(fact_7797_verit__sko__forall_H,axiom,
    ! [A: $tType,P2: fun(A,bool),A6: bool] :
      ( ( pp(aa(A,bool,P2,fChoice(A,aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2))))
      <=> pp(A6) )
     => ( ! [X_12: A] : pp(aa(A,bool,P2,X_12))
      <=> pp(A6) ) ) ).

% verit_sko_forall'
tff(fact_7798_verit__sko__forall,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ! [X_12: A] : pp(aa(A,bool,P2,X_12))
    <=> pp(aa(A,bool,P2,fChoice(A,aTP_Lamp_dg(fun(A,bool),fun(A,bool),P2)))) ) ).

% verit_sko_forall
tff(fact_7799_verit__sko__ex_H,axiom,
    ! [A: $tType,P2: fun(A,bool),A6: bool] :
      ( ( pp(aa(A,bool,P2,fChoice(A,P2)))
      <=> pp(A6) )
     => ( ? [X_12: A] : pp(aa(A,bool,P2,X_12))
      <=> pp(A6) ) ) ).

% verit_sko_ex'
tff(fact_7800_equiv__Eps__preserves,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6))),A6)) ) ) ).

% equiv_Eps_preserves
tff(fact_7801_equiv__Eps__in,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6))),X6)) ) ) ).

% equiv_Eps_in
tff(fact_7802_toCard__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(B,B))] : bNF_Greatest_toCard(A,B,A6,R3) = fChoice(fun(A,B),bNF_Gr1419584066657907630d_pred(A,B,A6,R3)) ).

% toCard_def
tff(fact_7803_proj__Eps,axiom,
    ! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A6,R3)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R3)))
       => ( aa(A,set(A),equiv_proj(A,A,R3),fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6))) = X6 ) ) ) ).

% proj_Eps
tff(fact_7804_fromCard__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R3: set(product_prod(B,B)),K2: B] : bNF_Gr5436034075474128252omCard(A,B,A6,R3,K2) = fChoice(A,aa(B,fun(A,bool),aa(set(product_prod(B,B)),fun(B,fun(A,bool)),aTP_Lamp_asw(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,bool))),A6),R3),K2)) ).

% fromCard_def
tff(fact_7805_Un__csum,axiom,
    ! [A: $tType,A6: set(A),B5: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B5))),bNF_Cardinal_csum(A,A,bNF_Ca6860139660246222851ard_of(A,A6),bNF_Ca6860139660246222851ard_of(A,B5)))),bNF_Wellorder_ordLeq(A,sum_sum(A,A)))) ).

% Un_csum
tff(fact_7806_Eps__Opt__def,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
       => ( eps_Opt(A,P2) = aa(A,option(A),some(A),fChoice(A,P2)) ) )
      & ( ~ ? [X_1: A] : pp(aa(A,bool,P2,X_1))
       => ( eps_Opt(A,P2) = none(A) ) ) ) ).

% Eps_Opt_def
tff(fact_7807_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( lattic7623131987881927897min_on(A,B,F3,S) = fChoice(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_asx(set(A),fun(fun(A,B),fun(A,bool)),S),F3)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_7808_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R2,X) = fChoice(A,aa(B,fun(A,bool),aTP_Lamp_at(set(product_prod(B,A)),fun(B,fun(A,bool)),R2),X)) ).

% fun_of_rel_def
tff(fact_7809_arg__max__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: fun(B,A),P2: fun(B,bool)] : lattices_ord_arg_max(B,A,F3,P2) = fChoice(B,lattic501386751176901750rg_max(B,A,F3,P2)) ) ).

% arg_max_def
tff(fact_7810_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_pa(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% Eps_case_prod_eq
tff(fact_7811_card__of__def,axiom,
    ! [A: $tType,A6: set(A)] : bNF_Ca6860139660246222851ard_of(A,A6) = fChoice(set(product_prod(A,A)),bNF_Ca8970107618336181345der_on(A,A6)) ).

% card_of_def
tff(fact_7812_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P2) = fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_asy(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P2))) ).

% split_paired_Eps
tff(fact_7813_cardSuc__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : bNF_Ca8387033319878233205ardSuc(A,R3) = fChoice(set(product_prod(set(A),set(A))),bNF_Ca6246979054910435723ardSuc(A,R3)) ).

% cardSuc_def
tff(fact_7814_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2)) = fChoice(product_prod(A,B),aTP_Lamp_pb(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P2)) ).

% Eps_case_prod
tff(fact_7815_is__arg__max__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),X: A] :
          ( pp(aa(A,bool,lattic501386751176901750rg_max(A,B,F3,P2),X))
        <=> ( pp(aa(A,bool,P2,X))
            & ! [Y4: A] :
                ( pp(aa(A,bool,P2,Y4))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y4)),aa(A,B,F3,X))) ) ) ) ) ).

% is_arg_max_linorder
tff(fact_7816_is__arg__max__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: fun(B,A),P2: fun(B,bool),X: B] :
          ( pp(aa(B,bool,lattic501386751176901750rg_max(B,A,F3,P2),X))
        <=> ( pp(aa(B,bool,P2,X))
            & ~ ? [Y4: B] :
                  ( pp(aa(B,bool,P2,Y4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X)),aa(B,A,F3,Y4))) ) ) ) ) ).

% is_arg_max_def
tff(fact_7817_arg__min__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: fun(B,A),P2: fun(B,bool)] : lattices_ord_arg_min(B,A,F3,P2) = fChoice(B,lattic501386751177426532rg_min(B,A,F3,P2)) ) ).

% arg_min_def
tff(fact_7818_filterlim__INF__INF,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set(A),I5: set(B),F3: fun(D,C),F4: fun(B,filter(D)),G5: fun(A,filter(C))] :
      ( ! [M5: A] :
          ( pp(aa(set(A),bool,member(A,M5),J4))
         => ? [X5: B] :
              ( pp(aa(set(B),bool,member(B,X5),I5))
              & pp(aa(filter(C),bool,aa(filter(C),fun(filter(C),bool),ord_less_eq(filter(C)),filtermap(D,C,F3,aa(B,filter(D),F4,X5))),aa(A,filter(C),G5,M5))) ) )
     => filterlim(D,C,F3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),G5),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),F4),I5))) ) ).

% filterlim_INF_INF
tff(fact_7819_filtermap__id_H,axiom,
    ! [A: $tType,X5: filter(A)] : filtermap(A,A,aTP_Lamp_cc(A,A),X5) = X5 ).

% filtermap_id'
tff(fact_7820_filtermap__inf,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),F14: filter(B),F23: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F14),F23))),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),filtermap(B,A,F3,F14)),filtermap(B,A,F3,F23)))) ).

% filtermap_inf
tff(fact_7821_filtermap__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),F4: fun(C,filter(B)),B5: set(C)] : filtermap(B,A,F3,aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B5))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_asz(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F3),F4)),B5)) ).

% filtermap_SUP
tff(fact_7822_filtermap__sup,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),F14: filter(B),F23: filter(B)] : filtermap(B,A,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F14),F23)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),filtermap(B,A,F3,F14)),filtermap(B,A,F3,F23)) ).

% filtermap_sup
tff(fact_7823_filtermap__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),F4: filter(C)] : filtermap(B,A,F3,filtermap(C,B,G3,F4)) = filtermap(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_vk(fun(B,A),fun(fun(C,B),fun(C,A)),F3),G3),F4) ).

% filtermap_filtermap
tff(fact_7824_filtermap__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filtermap(A,A,aTP_Lamp_cc(A,A),F4) = F4 ).

% filtermap_ident
tff(fact_7825_filterlim__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,B),F14: filter(B),G3: fun(C,A),F23: filter(C)] :
      ( filterlim(A,B,F3,F14,filtermap(C,A,G3,F23))
    <=> filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aob(fun(A,B),fun(fun(C,A),fun(C,B)),F3),G3),F14,F23) ) ).

% filterlim_filtermap
tff(fact_7826_filterlim__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F23: filter(B),F14: filter(A)] :
      ( filterlim(A,B,F3,F23,F14)
    <=> pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(A,B,F3,F14)),F23)) ) ).

% filterlim_def
tff(fact_7827_filtermap__mono,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),F6: filter(A),F3: fun(A,B)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F6))
     => pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(A,B,F3,F4)),filtermap(A,B,F3,F6))) ) ).

% filtermap_mono
tff(fact_7828_is__arg__min__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),X: A] :
          ( pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F3,P2),X))
        <=> ( pp(aa(A,bool,P2,X))
            & ! [Y4: A] :
                ( pp(aa(A,bool,P2,Y4))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y4))) ) ) ) ) ).

% is_arg_min_linorder
tff(fact_7829_is__arg__min__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),X: A,Y: A] :
          ( pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F3,P2),X))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X)))
           => ( pp(aa(A,bool,P2,Y))
             => pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F3,P2),Y)) ) ) ) ) ).

% is_arg_min_antimono
tff(fact_7830_filtermap__fun__inverse,axiom,
    ! [B: $tType,A: $tType,G3: fun(A,B),F4: filter(B),G5: filter(A),F3: fun(B,A)] :
      ( filterlim(A,B,G3,F4,G5)
     => ( filterlim(B,A,F3,G5,F4)
       => ( eventually(A,aa(fun(B,A),fun(A,bool),aTP_Lamp_ata(fun(A,B),fun(fun(B,A),fun(A,bool)),G3),F3),G5)
         => ( filtermap(B,A,F3,F4) = G5 ) ) ) ) ).

% filtermap_fun_inverse
tff(fact_7831_eventually__filtermap,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),F3: fun(B,A),F4: filter(B)] :
      ( eventually(A,P2,filtermap(B,A,F3,F4))
    <=> eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqc(fun(A,bool),fun(fun(B,A),fun(B,bool)),P2),F3),F4) ) ).

% eventually_filtermap
tff(fact_7832_filtermap__le__iff__le__filtercomap,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),F4: filter(B),G5: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F3,F4)),G5))
    <=> pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F4),filtercomap(B,A,F3,G5))) ) ).

% filtermap_le_iff_le_filtercomap
tff(fact_7833_filtermap__filtercomap,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),F4: filter(A)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F3,filtercomap(B,A,F3,F4))),F4)) ).

% filtermap_filtercomap
tff(fact_7834_filtercomap__filtermap,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),F3: fun(A,B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),filtercomap(A,B,F3,filtermap(A,B,F3,F4)))) ).

% filtercomap_filtermap
tff(fact_7835_is__arg__min__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: fun(B,A),P2: fun(B,bool),X: B] :
          ( pp(aa(B,bool,lattic501386751177426532rg_min(B,A,F3,P2),X))
        <=> ( pp(aa(B,bool,P2,X))
            & ~ ? [Y4: B] :
                  ( pp(aa(B,bool,P2,Y4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,Y4)),aa(B,A,F3,X))) ) ) ) ) ).

% is_arg_min_def
tff(fact_7836_filtermap__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(A)] : filtermap(A,B,F3,F4) = abs_filter(B,aa(filter(A),fun(fun(B,bool),bool),aTP_Lamp_atb(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),F3),F4)) ).

% filtermap_def
tff(fact_7837_filtermap__mono__strong,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(A),G5: filter(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(A,B,F3,F4)),filtermap(A,B,F3,G5)))
      <=> pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),G5)) ) ) ).

% filtermap_mono_strong
tff(fact_7838_is__arg__min__arg__min__nat,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P2,X))
     => pp(aa(A,bool,lattic501386751177426532rg_min(A,nat,M,P2),lattices_ord_arg_min(A,nat,M,P2))) ) ).

% is_arg_min_arg_min_nat
tff(fact_7839_filtermap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),F4: fun(C,filter(B)),B5: set(C)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F3,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F4),B5)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_asz(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F3),F4)),B5)))) ).

% filtermap_INF
tff(fact_7840_ex__is__arg__min__if__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S))
         => ( ( S != bot_bot(set(A)) )
           => ? [X_1: A] : pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F3,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),S)),X_1)) ) ) ) ).

% ex_is_arg_min_if_finite
tff(fact_7841_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),X: B] : prod_filter(A,B,F4,principal(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_oz(B,fun(A,product_prod(A,B))),X),F4) ).

% prod_filter_principal_singleton2
tff(fact_7842_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one))))),N))
         => ( unique5772411509450598832har_of(A,aa(A,A,bit_se2584673776208193580ke_bit(A,N),M)) = unique5772411509450598832har_of(A,M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_7843_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),F4: filter(C)] : pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_afc(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3),F4)),prod_filter(A,B,filtermap(C,A,F3,F4),filtermap(C,B,G3,F4)))) ).

% filtermap_Pair
tff(fact_7844_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: filter(A),G5: filter(B),H11: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G5),H11) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_atd(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F4,prod_filter(B,C,G5,H11))) ).

% prod_filter_assoc
tff(fact_7845_filtermap__snd__prod__filter,axiom,
    ! [B: $tType,A: $tType,A6: filter(B),B5: filter(A)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(product_prod(B,A),A,product_snd(B,A),prod_filter(B,A,A6,B5))),B5)) ).

% filtermap_snd_prod_filter
tff(fact_7846_filtermap__fst__prod__filter,axiom,
    ! [B: $tType,A: $tType,A6: filter(A),B5: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(product_prod(A,B),A,product_fst(A,B),prod_filter(A,B,A6,B5))),A6)) ).

% filtermap_fst_prod_filter
tff(fact_7847_prod__filter__mono,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F6: filter(A),G5: filter(B),G8: filter(B)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F4),F6))
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),G5),G8))
       => pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,F4,G5)),prod_filter(A,B,F6,G8))) ) ) ).

% prod_filter_mono
tff(fact_7848_prod__filter__mono__iff,axiom,
    ! [A: $tType,B: $tType,A6: filter(A),B5: filter(B),C4: filter(A),D5: filter(B)] :
      ( ( A6 != bot_bot(filter(A)) )
     => ( ( B5 != bot_bot(filter(B)) )
       => ( pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,A6,B5)),prod_filter(A,B,C4,D5)))
        <=> ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),A6),C4))
            & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),B5),D5)) ) ) ) ) ).

% prod_filter_mono_iff
tff(fact_7849_principal__prod__principal,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B5: set(B)] : prod_filter(A,B,principal(A,A6),principal(B,B5)) = principal(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aaz(set(B),fun(A,set(B)),B5))) ).

% principal_prod_principal
tff(fact_7850_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,B),G5: filter(B),F4: filter(A),G3: fun(A,C),H11: filter(C)] :
      ( filterlim(A,B,F3,G5,F4)
     => ( filterlim(A,C,G3,H11,F4)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ate(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3),prod_filter(B,C,G5,H11),F4) ) ) ).

% filterlim_Pair
tff(fact_7851_eventually__prod__same,axiom,
    ! [A: $tType,P2: fun(product_prod(A,A),bool),F4: filter(A)] :
      ( eventually(product_prod(A,A),P2,prod_filter(A,A,F4,F4))
    <=> ? [Q9: fun(A,bool)] :
          ( eventually(A,Q9,F4)
          & ! [X4: A,Y4: A] :
              ( pp(aa(A,bool,Q9,X4))
             => ( pp(aa(A,bool,Q9,Y4))
               => pp(aa(product_prod(A,A),bool,P2,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4))) ) ) ) ) ).

% eventually_prod_same
tff(fact_7852_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P2: fun(product_prod(A,B),bool),F4: filter(A),G5: filter(B)] :
      ( eventually(product_prod(A,B),P2,prod_filter(A,B,F4,G5))
    <=> ? [Pf: fun(A,bool),Pg: fun(B,bool)] :
          ( eventually(A,Pf,F4)
          & eventually(B,Pg,G5)
          & ! [X4: A,Y4: B] :
              ( pp(aa(A,bool,Pf,X4))
             => ( pp(aa(B,bool,Pg,Y4))
               => pp(aa(product_prod(A,B),bool,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4))) ) ) ) ) ).

% eventually_prod_filter
tff(fact_7853_prod__filter__def,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),G5: filter(B)] : prod_filter(A,B,F4,G5) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,bool),fun(B,bool))),set(filter(product_prod(A,B))),image2(product_prod(fun(A,bool),fun(B,bool)),filter(product_prod(A,B)),aa(fun(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))),fun(product_prod(fun(A,bool),fun(B,bool)),filter(product_prod(A,B))),product_case_prod(fun(A,bool),fun(B,bool),filter(product_prod(A,B))),aTP_Lamp_atf(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))))),aa(fun(product_prod(fun(A,bool),fun(B,bool)),bool),set(product_prod(fun(A,bool),fun(B,bool))),collect(product_prod(fun(A,bool),fun(B,bool))),aa(fun(fun(A,bool),fun(fun(B,bool),bool)),fun(product_prod(fun(A,bool),fun(B,bool)),bool),product_case_prod(fun(A,bool),fun(B,bool),bool),aa(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool)),aTP_Lamp_atg(filter(A),fun(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool))),F4),G5))))) ).

% prod_filter_def
tff(fact_7854_le__prod__filterI,axiom,
    ! [A: $tType,B: $tType,F4: filter(product_prod(A,B)),A6: filter(A),B5: filter(B)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(product_prod(A,B),A,product_fst(A,B),F4)),A6))
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(product_prod(A,B),B,product_snd(A,B),F4)),B5))
       => pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),F4),prod_filter(A,B,A6,B5))) ) ) ).

% le_prod_filterI
tff(fact_7855_eventually__prod__sequentially,axiom,
    ! [P2: fun(product_prod(nat,nat),bool)] :
      ( eventually(product_prod(nat,nat),P2,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N13: nat] :
        ! [M3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N13),M3))
         => ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N13),N3))
             => pp(aa(product_prod(nat,nat),bool,P2,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N3),M3))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_7856_eventually__prodI,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),F4: filter(A),Q2: fun(B,bool),G5: filter(B)] :
      ( eventually(A,P2,F4)
     => ( eventually(B,Q2,G5)
       => eventually(product_prod(A,B),aa(fun(B,bool),fun(product_prod(A,B),bool),aTP_Lamp_ath(fun(A,bool),fun(fun(B,bool),fun(product_prod(A,B),bool)),P2),Q2),prod_filter(A,B,F4,G5)) ) ) ).

% eventually_prodI
tff(fact_7857_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A6: filter(A),P2: fun(B,bool),B5: filter(B)] :
      ( ( A6 != bot_bot(filter(A)) )
     => ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ati(fun(B,bool),fun(A,fun(B,bool)),P2)),prod_filter(A,B,A6,B5))
      <=> eventually(B,P2,B5) ) ) ).

% eventually_prod2
tff(fact_7858_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B5: filter(A),P2: fun(B,bool),A6: filter(B)] :
      ( ( B5 != bot_bot(filter(A)) )
     => ( eventually(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),aTP_Lamp_atj(fun(B,bool),fun(B,fun(A,bool)),P2)),prod_filter(B,A,A6,B5))
      <=> eventually(B,P2,A6) ) ) ).

% eventually_prod1
tff(fact_7859_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I5: set(A),J4: set(B),A6: fun(A,filter(C)),B5: fun(B,filter(D))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ( J4 != bot_bot(set(B)) )
       => ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),A6),I5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B5),J4))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_atl(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A6),B5)),I5)) ) ) ) ).

% prod_filter_INF
tff(fact_7860_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A6: fun(A,filter(B)),B5: filter(C)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),A6),I5)),B5) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_atm(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A6),B5)),I5)) ) ) ).

% prod_filter_INF1
tff(fact_7861_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J4: set(A),A6: filter(B),B5: fun(A,filter(C))] :
      ( ( J4 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A6,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B5),J4))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_atn(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A6),B5)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_7862_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,F4: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))),F4) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),F4) ).

% prod_filter_principal_singleton
tff(fact_7863_sym__trans__comp__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( sym(A,R3)
     => ( trans(A,R3)
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,converse(A,A,R3),R3)),R3)) ) ) ).

% sym_trans_comp_subset
tff(fact_7864_snga__assn__raw_Opelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: array(A),Xa2: list(A),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa2),Xb2))
         => ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
           => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                  ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
                 => ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
                   => ( ( array_get(A,H2,X) = Xa2 )
                      & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H2))) ) ) ) ) ) ) ).

% snga_assn_raw.pelims(3)
tff(fact_7865_sym__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( sym(A,R3)
    <=> ! [X4: A,Y4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4)),R3))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4)),R3)) ) ) ).

% sym_def
tff(fact_7866_symI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)),R3))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R3)) )
     => sym(A,R3) ) ).

% symI
tff(fact_7867_symE,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B2: A,A3: A] :
      ( sym(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ).

% symE
tff(fact_7868_symD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),B2: A,A3: A] :
      ( sym(A,R3)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R3))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3)) ) ) ).

% symD
tff(fact_7869_sym__Un__converse,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : sym(A,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))),R3),converse(A,A,R3))) ).

% sym_Un_converse
tff(fact_7870_sym__Un,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( sym(A,R3)
     => ( sym(A,S2)
       => sym(A,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))),R3),S2)) ) ) ).

% sym_Un
tff(fact_7871_relH__array,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [As: set(nat),H: heap_ext(product_unit),H5: heap_ext(product_unit),R3: array(A)] :
          ( relH(As,H,H5)
         => ( pp(aa(set(nat),bool,member(nat,addr_of_array(A,R3)),As))
           => ( array_get(A,H,R3) = array_get(A,H5,R3) ) ) ) ) ).

% relH_array
tff(fact_7872_Array__Time_Opresent__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [H: heap_ext(product_unit),A3: array(A)] :
          ( array_present(A,H,A3)
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,A3)),lim(product_unit,H))) ) ) ).

% Array_Time.present_def
tff(fact_7873_snga__assn__raw_Osimps,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: array(A),X: list(A),H: heap_ext(product_unit),As: set(nat)] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,R3,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
        <=> ( ( array_get(A,H,R3) = X )
            & ( As = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_array(A,R3)),bot_bot(set(nat))) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,R3)),lim(product_unit,H))) ) ) ) ).

% snga_assn_raw.simps
tff(fact_7874_snga__assn__raw_Oelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: array(A),Xa2: list(A),Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
          ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa2),Xb2))
          <=> pp(Y) )
         => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
               => ( pp(Y)
                <=> ~ ( ( array_get(A,H2,X) = Xa2 )
                      & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H2))) ) ) ) ) ) ).

% snga_assn_raw.elims(1)
tff(fact_7875_snga__assn__raw_Oelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: array(A),Xa2: list(A),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa2),Xb2))
         => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
               => ~ ( ( array_get(A,H2,X) = Xa2 )
                    & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H2))) ) ) ) ) ).

% snga_assn_raw.elims(2)
tff(fact_7876_snga__assn__raw_Oelims_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: array(A),Xa2: list(A),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa2),Xb2))
         => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
               => ( ( array_get(A,H2,X) = Xa2 )
                  & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H2))) ) ) ) ) ).

% snga_assn_raw.elims(3)
tff(fact_7877_snga__assn__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: array(A),A3: list(A)] : snga_assn(A,R3,A3) = abs_assn(snga_assn_raw(A,R3,A3)) ) ).

% snga_assn_def
tff(fact_7878_snga__assn__raw_Opelims_I1_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: array(A),Xa2: list(A),Xb2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
          ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa2),Xb2))
          <=> pp(Y) )
         => ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
           => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                  ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
                 => ( ( pp(Y)
                    <=> ( ( array_get(A,H2,X) = Xa2 )
                        & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H2))) ) )
                   => ~ pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))))) ) ) ) ) ) ).

% snga_assn_raw.pelims(1)
tff(fact_7879_snga__assn__raw_Opelims_I2_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: array(A),Xa2: list(A),Xb2: product_prod(heap_ext(product_unit),set(nat))] :
          ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,snga_assn_raw(A,X,Xa2),Xb2))
         => ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa2),Xb2))))
           => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
                  ( ( Xb2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
                 => ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))))
                   => ~ ( ( array_get(A,H2,X) = Xa2 )
                        & ( As2 = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),addr_of_array(A,X)),bot_bot(set(nat))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X)),lim(product_unit,H2))) ) ) ) ) ) ) ).

% snga_assn_raw.pelims(2)
tff(fact_7880_relH__set__array,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [R3: array(A),As: set(nat),H: heap_ext(product_unit),X: list(A)] :
          ( ~ pp(aa(set(nat),bool,member(nat,addr_of_array(A,R3)),As))
         => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
           => relH(As,H,array_set(A,R3,X,H)) ) ) ) ).

% relH_set_array
tff(fact_7881_subset__singleton__iff__Uniq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ? [A9: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A9),bot_bot(set(A)))))
    <=> uniq(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)) ) ).

% subset_singleton_iff_Uniq
tff(fact_7882_left__unique__iff,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] :
      ( left_unique(A,B,R2)
    <=> ! [Z2: B] : uniq(A,aa(B,fun(A,bool),aTP_Lamp_adr(fun(A,fun(B,bool)),fun(B,fun(A,bool)),R2),Z2)) ) ).

% left_unique_iff
tff(fact_7883_the1__equality_H,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( uniq(A,P2)
     => ( pp(aa(A,bool,P2,A3))
       => ( the(A,P2) = A3 ) ) ) ).

% the1_equality'
tff(fact_7884_alt__ex1E_H,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P2,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P2,Y3))
             => ( Y3 = X5 ) ) )
     => ~ ( ? [X_1: A] : pp(aa(A,bool,P2,X_1))
         => ~ uniq(A,P2) ) ) ).

% alt_ex1E'
tff(fact_7885_ex1__iff__ex__Uniq,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(A,bool,P2,X4))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P2,Y4))
             => ( Y4 = X4 ) ) )
    <=> ( ? [X_12: A] : pp(aa(A,bool,P2,X_12))
        & uniq(A,P2) ) ) ).

% ex1_iff_ex_Uniq
tff(fact_7886_inj__on__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A)] :
      ( inj_on(A,B,F3,A6)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A6))
         => uniq(A,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_ato(fun(A,B),fun(set(A),fun(A,fun(A,bool))),F3),A6),X4)) ) ) ).

% inj_on_iff_Uniq
tff(fact_7887_bi__unique__iff,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,R2)
    <=> ( ! [Z2: B] : uniq(A,aa(B,fun(A,bool),aTP_Lamp_adr(fun(A,fun(B,bool)),fun(B,fun(A,bool)),R2),Z2))
        & ! [Z2: A] : uniq(B,aa(A,fun(B,bool),R2,Z2)) ) ) ).

% bi_unique_iff
tff(fact_7888_right__unique__iff,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( right_unique(A,B,R2)
    <=> ! [Z2: A] : uniq(B,aa(A,fun(B,bool),R2,Z2)) ) ).

% right_unique_iff
tff(fact_7889_pairwise__disjnt__iff,axiom,
    ! [A: $tType,A18: set(set(A))] :
      ( pairwise(set(A),disjnt(A),A18)
    <=> ! [X4: A] : uniq(set(A),aa(A,fun(set(A),bool),aTP_Lamp_atp(set(set(A)),fun(A,fun(set(A),bool)),A18),X4)) ) ).

% pairwise_disjnt_iff
tff(fact_7890_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : uniq(list(A),aTP_Lamp_atq(set(A),fun(list(A),bool),A6)) ) ).

% strict_sorted_equal_Uniq
tff(fact_7891_Array__Time_Oalloc__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A),H: heap_ext(product_unit)] : array_alloc(A,Xs,H) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),array2(A,lim(product_unit,H))),array_set(A,array2(A,lim(product_unit,H)),Xs,lim_update(product_unit,aTP_Lamp_atr(heap_ext(product_unit),fun(nat,nat),H),H))) ) ).

% Array_Time.alloc_def
tff(fact_7892_prod__decode__triangle__add,axiom,
    ! [K2: nat,M: nat] : nat_prod_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K2)),M)) = nat_prod_decode_aux(K2,M) ).

% prod_decode_triangle_add
tff(fact_7893_Ref__Time_Oalloc__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [X: A,H: heap_ext(product_unit)] : ref_alloc(A,X,H) = aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),ref2(A,lim(product_unit,H))),ref_set(A,ref2(A,lim(product_unit,H)),X,lim_update(product_unit,aTP_Lamp_atr(heap_ext(product_unit),fun(nat,nat),H),H))) ) ).

% Ref_Time.alloc_def
tff(fact_7894_unfold__congs_I3_J,axiom,
    ! [Z11: $tType,R3: heap_ext(Z11),R6: heap_ext(Z11),V6: nat,F3: fun(nat,nat),F5: fun(nat,nat)] :
      ( ( R3 = R6 )
     => ( ( lim(Z11,R6) = V6 )
       => ( ! [V3: nat] :
              ( ( V3 = V6 )
             => ( aa(nat,nat,F3,V3) = aa(nat,nat,F5,V3) ) )
         => ( lim_update(Z11,F3,R3) = lim_update(Z11,F5,R6) ) ) ) ) ).

% unfold_congs(3)
tff(fact_7895_fold__congs_I3_J,axiom,
    ! [Z11: $tType,R3: heap_ext(Z11),R6: heap_ext(Z11),V6: nat,F3: fun(nat,nat),F5: fun(nat,nat)] :
      ( ( R3 = R6 )
     => ( ( lim(Z11,R6) = V6 )
       => ( ! [V3: nat] :
              ( ( V6 = V3 )
             => ( aa(nat,nat,F3,V3) = aa(nat,nat,F5,V3) ) )
         => ( lim_update(Z11,F3,R3) = lim_update(Z11,F5,R6) ) ) ) ) ).

% fold_congs(3)
tff(fact_7896_list__decode_Oelims,axiom,
    ! [X: nat,Y: list(nat)] :
      ( ( nat_list_decode(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != nil(nat) ) )
       => ~ ! [N4: nat] :
              ( ( X = aa(nat,nat,suc,N4) )
             => ( Y != aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ats(nat,fun(nat,list(nat)))),nat_prod_decode(N4)) ) ) ) ) ).

% list_decode.elims
tff(fact_7897_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : nat_list_decode(aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ats(nat,fun(nat,list(nat)))),nat_prod_decode(N)) ).

% list_decode.simps(2)
tff(fact_7898_list__decode_Opelims,axiom,
    ! [X: nat,Y: list(nat)] :
      ( ( nat_list_decode(X) = Y )
     => ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),X))
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = nil(nat) )
             => ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat))) ) )
         => ~ ! [N4: nat] :
                ( ( X = aa(nat,nat,suc,N4) )
               => ( ( Y = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ats(nat,fun(nat,list(nat)))),nat_prod_decode(N4)) )
                 => ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N4))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_7899_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N)))
     => ( nat_list_decode(aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ats(nat,fun(nat,list(nat)))),nat_prod_decode(N)) ) ) ).

% list_decode.psimps(2)
tff(fact_7900_gen__length__def,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : gen_length(A,N,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)) ).

% gen_length_def
tff(fact_7901_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A))))
      <=> ? [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
            & ( aa(nat,A,semiring_1_of_nat(A),N3) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7902_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
           => ( aa(nat,A,semiring_1_of_nat(A),N3) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7903_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C3))
         => ( ( aa(nat,A,semiring_1_of_nat(A),C3) = zero_zero(A) )
           => ( ! [X3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X3))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),C3))
                   => ( aa(nat,A,semiring_1_of_nat(A),X3) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C3 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7904_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : bit_se4197421643247451524op_bit(A,M,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% drop_bit_exp_eq
tff(fact_7905_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% bit_minus_2_iff
tff(fact_7906_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I2: nat,J2: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,I2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
           => pp(bit_se6407376104438227557le_bit(A,Tyrep,J2)) ) ) ) ).

% possible_bit_less_imp
tff(fact_7907_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,M)),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% bit_mask_iff
tff(fact_7908_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se4730199178511100633sh_bit(A,M,A3)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% bit_push_bit_iff
tff(fact_7909_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7910_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),M)),one_one(A))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7911_subset__mset_Osum__pos2,axiom,
    ! [A: $tType,B: $tType,I5: set(B),I2: B,F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),I5))
     => ( pp(aa(set(B),bool,member(B,I2),I5))
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I2)))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,member(B,I3),I5))
               => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I3))) )
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,I5))) ) ) ) ) ).

% subset_mset.sum_pos2
tff(fact_7912_card__def,axiom,
    ! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_att(B,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_7913_subset__mset_Oadd__less__cancel__right,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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)),B2),C3)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2)) ) ).

% subset_mset.add_less_cancel_right
tff(fact_7914_subset__mset_Oadd__less__cancel__left,axiom,
    ! [A: $tType,C3: multiset(A),A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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),B2)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2)) ) ).

% subset_mset.add_less_cancel_left
tff(fact_7915_subset__mset_Oless__add__same__cancel2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),B2)) ) ).

% subset_mset.less_add_same_cancel2
tff(fact_7916_subset__mset_Oless__add__same__cancel1,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),B2)) ) ).

% subset_mset.less_add_same_cancel1
tff(fact_7917_subset__mset_Oadd__less__same__cancel2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),B2))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),zero_zero(multiset(A)))) ) ).

% subset_mset.add_less_same_cancel2
tff(fact_7918_subset__mset_Oadd__less__same__cancel1,axiom,
    ! [A: $tType,B2: multiset(A),A3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3)),B2))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),zero_zero(multiset(A)))) ) ).

% subset_mset.add_less_same_cancel1
tff(fact_7919_subset__mset_Olexordp_Omono,axiom,
    ! [A: $tType] : pp(aa(fun(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))),bool,order_mono(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))),aTP_Lamp_atu(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))))) ).

% subset_mset.lexordp.mono
tff(fact_7920_subset__mset_Olexordp__def,axiom,
    ! [A: $tType] : lexordp2(multiset(A),subset_mset(A)) = complete_lattice_lfp(fun(list(multiset(A)),fun(list(multiset(A)),bool)),aTP_Lamp_atu(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool)))) ).

% subset_mset.lexordp_def
tff(fact_7921_size__psubset,axiom,
    ! [A: $tType,M6: multiset(A),M9: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M6),M9))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),M6)),aa(multiset(A),nat,size_size(multiset(A)),M9)))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),M6),M9)) ) ) ).

% size_psubset
tff(fact_7922_subset__mset_Oadd__neg__nonpos,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),zero_zero(multiset(A))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),zero_zero(multiset(A))))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),zero_zero(multiset(A)))) ) ) ).

% subset_mset.add_neg_nonpos
tff(fact_7923_subset__mset_Oadd__nonneg__pos,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),B2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2))) ) ) ).

% subset_mset.add_nonneg_pos
tff(fact_7924_subset__mset_Oadd__nonpos__neg,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),zero_zero(multiset(A))))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),zero_zero(multiset(A)))) ) ) ).

% subset_mset.add_nonpos_neg
tff(fact_7925_subset__mset_Oadd__pos__nonneg,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2))) ) ) ).

% subset_mset.add_pos_nonneg
tff(fact_7926_subset__mset_Oadd__strict__increasing,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),C3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3))) ) ) ).

% subset_mset.add_strict_increasing
tff(fact_7927_subset__mset_Oadd__strict__increasing2,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),C3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3))) ) ) ).

% subset_mset.add_strict_increasing2
tff(fact_7928_subset__mset_OSup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] : lattic4895041142388067077er_set(multiset(A),union_mset(A),aTP_Lamp_atv(multiset(A),fun(multiset(A),bool)),aTP_Lamp_atw(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.Sup_fin.semilattice_order_set_axioms
tff(fact_7929_subset__mset_Oasymp__greater,axiom,
    ! [A: $tType] : asymp(multiset(A),aTP_Lamp_atw(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.asymp_greater
tff(fact_7930_subset__mset_Oadd__strict__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A),D3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),C3),D3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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)),B2),D3))) ) ) ).

% subset_mset.add_strict_mono
tff(fact_7931_subset__mset_Oadd__strict__left__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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),B2))) ) ).

% subset_mset.add_strict_left_mono
tff(fact_7932_subset__mset_Oadd__strict__right__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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)),B2),C3))) ) ).

% subset_mset.add_strict_right_mono
tff(fact_7933_subset__mset_Oadd__less__imp__less__left,axiom,
    ! [A: $tType,C3: multiset(A),A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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),B2)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2)) ) ).

% subset_mset.add_less_imp_less_left
tff(fact_7934_subset__mset_Oadd__less__imp__less__right,axiom,
    ! [A: $tType,A3: multiset(A),C3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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)),B2),C3)))
     => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2)) ) ).

% subset_mset.add_less_imp_less_right
tff(fact_7935_subset__mset_Oadd__less__le__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A),D3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C3),D3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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)),B2),D3))) ) ) ).

% subset_mset.add_less_le_mono
tff(fact_7936_subset__mset_Oadd__le__less__mono,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A),D3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),C3),D3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_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)),B2),D3))) ) ) ).

% subset_mset.add_le_less_mono
tff(fact_7937_subset__mset_Opos__add__strict,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),C3))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3))) ) ) ).

% subset_mset.pos_add_strict
tff(fact_7938_subset__mset_Oadd__pos__pos,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),A3))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),B2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2))) ) ) ).

% subset_mset.add_pos_pos
tff(fact_7939_subset__mset_Oadd__neg__neg,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),zero_zero(multiset(A))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),B2),zero_zero(multiset(A))))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),zero_zero(multiset(A)))) ) ) ).

% subset_mset.add_neg_neg
tff(fact_7940_subset__mset_OlessE,axiom,
    ! [A: $tType,A3: multiset(A),B2: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A3),B2))
     => ~ ! [C2: multiset(A)] :
            ( ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2) )
           => ( C2 = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.lessE
tff(fact_7941_subset__mset_Oordering__top__axioms,axiom,
    ! [A: $tType] : ordering_top(multiset(A),aTP_Lamp_atv(multiset(A),fun(multiset(A),bool)),aTP_Lamp_atw(multiset(A),fun(multiset(A),bool)),zero_zero(multiset(A))) ).

% subset_mset.ordering_top_axioms
tff(fact_7942_mset__subset__size,axiom,
    ! [A: $tType,A6: multiset(A),B5: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),A6),B5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(multiset(A),nat,size_size(multiset(A)),A6)),aa(multiset(A),nat,size_size(multiset(A)),B5))) ) ).

% mset_subset_size
tff(fact_7943_subset__mset_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N: nat,M: nat] :
      ( ! [N4: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N4)),aa(nat,multiset(A),F3,aa(nat,nat,suc,N4))))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N)),aa(nat,multiset(A),F3,M)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).

% subset_mset.lift_Suc_mono_less_iff
tff(fact_7944_subset__mset_Olift__Suc__mono__less,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),N: nat,N2: nat] :
      ( ! [N4: nat] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N4)),aa(nat,multiset(A),F3,aa(nat,nat,suc,N4))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N2))
       => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(nat,multiset(A),F3,N)),aa(nat,multiset(A),F3,N2))) ) ) ).

% subset_mset.lift_Suc_mono_less
tff(fact_7945_wf__subset__mset__rel,axiom,
    ! [A: $tType] : wf(multiset(A),aa(fun(product_prod(multiset(A),multiset(A)),bool),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),bool)),fun(product_prod(multiset(A),multiset(A)),bool),product_case_prod(multiset(A),multiset(A),bool),subset_mset(A)))) ).

% wf_subset_mset_rel
tff(fact_7946_subset__mset_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] : semila1105856199041335345_order(multiset(A),union_mset(A),zero_zero(multiset(A)),aTP_Lamp_atv(multiset(A),fun(multiset(A),bool)),aTP_Lamp_atw(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.semilattice_neutr_order_axioms
tff(fact_7947_mset__subset__add__iff1,axiom,
    ! [A: $tType,J2: nat,I2: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J2),U)),N)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M)),N)) ) ) ).

% mset_subset_add_iff1
tff(fact_7948_mset__subset__add__iff2,axiom,
    ! [A: $tType,I2: nat,J2: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),I2),U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),J2),U)),N)))
      <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(nat,fun(multiset(A),multiset(A)),repeat_mset(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N))) ) ) ).

% mset_subset_add_iff2
tff(fact_7949_subset__mset_OcSUP__lessD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,multiset(A)),A6: set(B),Y: multiset(A),I2: B] :
      ( pp(aa(set(multiset(A)),bool,condit8047198070973881523_above(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(set(multiset(A)),multiset(A),complete_Sup_Sup(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6))),Y))
       => ( pp(aa(set(B),bool,member(B,I2),A6))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(B,multiset(A),F3,I2)),Y)) ) ) ) ).

% subset_mset.cSUP_lessD
tff(fact_7950_subset__mset_Oless__cINF__D,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,multiset(A)),A6: set(B),Y: multiset(A),I2: B] :
      ( pp(aa(set(multiset(A)),bool,condit8119078960628432327_below(multiset(A),subseteq_mset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6)))
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Y),aa(set(multiset(A)),multiset(A),complete_Inf_Inf(multiset(A)),aa(set(B),set(multiset(A)),image2(B,multiset(A),F3),A6))))
       => ( pp(aa(set(B),bool,member(B,I2),A6))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Y),aa(B,multiset(A),F3,I2))) ) ) ) ).

% subset_mset.less_cINF_D
tff(fact_7951_subset__mset_Osum__pos,axiom,
    ! [A: $tType,B: $tType,I5: set(B),F3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),I5))
     => ( ( I5 != bot_bot(set(B)) )
       => ( ! [I3: B] :
              ( pp(aa(set(B),bool,member(B,I3),I5))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F3,I3))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),zero_zero(multiset(A))),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,I5))) ) ) ) ).

% subset_mset.sum_pos
tff(fact_7952_subset__mset_Osum__strict__mono,axiom,
    ! [A: $tType,B: $tType,A6: set(B),F3: fun(B,multiset(A)),G3: fun(B,multiset(A))] :
      ( pp(aa(set(B),bool,finite_finite2(B),A6))
     => ( ( A6 != bot_bot(set(B)) )
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A6))
             => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),aa(B,multiset(A),F3,X3)),aa(B,multiset(A),G3,X3))) )
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F3,A6)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),G3,A6))) ) ) ) ).

% subset_mset.sum_strict_mono
tff(fact_7953_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),A6: set(A),X: A,Z4: B] :
      ( finite_folding_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(set(A),B,finite_folding_F(A,B,F3,Z4),A6) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_7954_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite_folding_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),B,finite_folding_F(A,B,F3,Z4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_7955_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_rb(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_7956_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite_folding_on(A,list(A),top_top(set(A)),linorder_insort_key(A,A,aTP_Lamp_pk(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_7957_folding__on_Oinsert,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite_folding_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ~ pp(aa(set(A),bool,member(A,X),A6))
           => ( aa(set(A),B,finite_folding_F(A,B,F3,Z4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z4),A6)) ) ) ) ) ) ).

% folding_on.insert
tff(fact_7958_folding__idem__on_Oinsert__idem,axiom,
    ! [B: $tType,A: $tType,S: set(A),F3: fun(A,fun(B,B)),X: A,A6: set(A),Z4: B] :
      ( finite1890593828518410140dem_on(A,B,S,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)),S))
       => ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),B,finite_folding_F(A,B,F3,Z4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A6)) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z4),A6)) ) ) ) ) ).

% folding_idem_on.insert_idem
tff(fact_7959_less__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A6)
     => pp(aa(fun(filter(B),fun(filter(B),bool)),bool,aa(fun(filter(A),fun(filter(A),bool)),fun(fun(filter(B),fun(filter(B),bool)),bool),bNF_rel_fun(filter(A),filter(B),fun(filter(A),bool),fun(filter(B),bool),rel_filter(A,B,A6),bNF_rel_fun(filter(A),filter(B),bool,bool,rel_filter(A,B,A6),fequal(bool))),ord_less(filter(A))),ord_less(filter(B)))) ) ).

% less_filter_parametric
tff(fact_7960_sup__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] : pp(aa(fun(filter(B),fun(filter(B),filter(B))),bool,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),bool),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A6),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A6),rel_filter(A,B,A6))),sup_sup(filter(A))),sup_sup(filter(B)))) ).

% sup_filter_parametric
tff(fact_7961_rel__filter__mono,axiom,
    ! [B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A6),B5))
     => pp(aa(fun(filter(A),fun(filter(B),bool)),bool,aa(fun(filter(A),fun(filter(B),bool)),fun(fun(filter(A),fun(filter(B),bool)),bool),ord_less_eq(fun(filter(A),fun(filter(B),bool))),rel_filter(A,B,A6)),rel_filter(A,B,B5))) ) ).

% rel_filter_mono
tff(fact_7962_le__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,A6)
     => pp(aa(fun(filter(B),fun(filter(B),bool)),bool,aa(fun(filter(A),fun(filter(A),bool)),fun(fun(filter(B),fun(filter(B),bool)),bool),bNF_rel_fun(filter(A),filter(B),fun(filter(A),bool),fun(filter(B),bool),rel_filter(A,B,A6),bNF_rel_fun(filter(A),filter(B),bool,bool,rel_filter(A,B,A6),fequal(bool))),ord_less_eq(filter(A))),ord_less_eq(filter(B)))) ) ).

% le_filter_parametric
tff(fact_7963_rel__filter_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),F4: filter(A),G5: filter(B)] :
      ( pp(aa(filter(B),bool,aa(filter(A),fun(filter(B),bool),rel_filter(A,B,R2),F4),G5))
     => ~ ! [Z8: filter(product_prod(A,B))] :
            ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2),Z8)
           => ( ( map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_fst(A,B),Z8) = F4 )
             => ( map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_snd(A,B),Z8) != G5 ) ) ) ) ).

% rel_filter.cases
tff(fact_7964_rel__filter_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),F4: filter(A),G5: filter(B)] :
      ( pp(aa(filter(B),bool,aa(filter(A),fun(filter(B),bool),rel_filter(A,B,R2),F4),G5))
    <=> ? [Z9: filter(product_prod(A,B))] :
          ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2),Z9)
          & ( map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_fst(A,B),Z9) = F4 )
          & ( map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_snd(A,B),Z9) = G5 ) ) ) ).

% rel_filter.simps
tff(fact_7965_rel__filter_Ointros,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),Z5: filter(product_prod(A,B)),F4: filter(A),G5: filter(B)] :
      ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2),Z5)
     => ( ( map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_fst(A,B),Z5) = F4 )
       => ( ( map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_snd(A,B),Z5) = G5 )
         => pp(aa(filter(B),bool,aa(filter(A),fun(filter(B),bool),rel_filter(A,B,R2),F4),G5)) ) ) ) ).

% rel_filter.intros
tff(fact_7966_partition__set,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A),Yes2: list(A),No2: list(A)] :
      ( ( partition(A,P2,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)),Yes2),No2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes2)),aa(list(A),set(A),set2(A),No2)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_7967_map__comp__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,option(C)),G3: fun(A,option(B)),X5: A] : map_comp(B,C,A,F3,G3,X5) = case_option(option(C),B,none(C),F3,aa(A,option(B),G3,X5)) ).

% map_comp_def
tff(fact_7968_map__comp__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,A: $tType,M22: fun(B,option(A)),K2: B,K10: A,M1: fun(A,option(C))] :
      ( ( aa(B,option(A),M22,K2) = aa(A,option(A),some(A),K10) )
     => ( map_comp(A,C,B,M1,M22,K2) = aa(A,option(C),M1,K10) ) ) ).

% map_comp_simps(2)
tff(fact_7969_map__comp__empty_I2_J,axiom,
    ! [C: $tType,B: $tType,D: $tType,M: fun(C,option(B)),X5: C] : map_comp(B,D,C,aTP_Lamp_atx(B,option(D)),M,X5) = none(D) ).

% map_comp_empty(2)
tff(fact_7970_map__comp__empty_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,M: fun(C,option(B)),X5: A] : map_comp(C,B,A,M,aTP_Lamp_mt(A,option(C)),X5) = none(B) ).

% map_comp_empty(1)
tff(fact_7971_map__comp__Some__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K2: C,V2: A] :
      ( ( map_comp(B,A,C,M1,M22,K2) = aa(A,option(A),some(A),V2) )
    <=> ? [K11: B] :
          ( ( aa(C,option(B),M22,K2) = aa(B,option(B),some(B),K11) )
          & ( aa(B,option(A),M1,K11) = aa(A,option(A),some(A),V2) ) ) ) ).

% map_comp_Some_iff
tff(fact_7972_map__comp__None__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K2: C] :
      ( ( map_comp(B,A,C,M1,M22,K2) = none(A) )
    <=> ( ( aa(C,option(B),M22,K2) = none(B) )
        | ? [K11: B] :
            ( ( aa(C,option(B),M22,K2) = aa(B,option(B),some(B),K11) )
            & ( aa(B,option(A),M1,K11) = none(A) ) ) ) ) ).

% map_comp_None_iff
tff(fact_7973_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,Xs: list(A)] : partition(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_aty(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P2),X)),partition(A,P2,Xs)) ).

% partition.simps(2)
tff(fact_7974_rel__fun__iff__geq__image2p,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R2: fun(A,fun(B,bool)),S: fun(C,fun(D,bool)),F3: fun(A,C),G3: fun(B,D)] :
      ( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,R2,S),F3),G3))
    <=> pp(aa(fun(C,fun(D,bool)),bool,aa(fun(C,fun(D,bool)),fun(fun(C,fun(D,bool)),bool),ord_less_eq(fun(C,fun(D,bool))),bNF_Greatest_image2p(A,C,B,D,F3,G3,R2)),S)) ) ).

% rel_fun_iff_geq_image2p
tff(fact_7975_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_atz(A,bool)) ) ) ).

% Nats_altdef1
tff(fact_7976_Nats__subset__Ints,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A))) ) ).

% Nats_subset_Ints
tff(fact_7977_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),semiring_1_Nats(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ) ).

% Nats_diff
tff(fact_7978_rel__fun__image2p,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R2: fun(A,fun(B,bool)),F3: fun(A,C),G3: fun(B,D)] : pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,R2,bNF_Greatest_image2p(A,C,B,D,F3,G3,R2)),F3),G3)) ).

% rel_fun_image2p
tff(fact_7979_Nats__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),field_char_0_Rats(A))) ) ).

% Nats_subset_Rats
tff(fact_7980_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,member(A,aa(nat,A,semiring_1_of_nat(A),N)),semiring_1_Nats(A))) ) ).

% of_nat_in_Nats
tff(fact_7981_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A,P2: fun(A,bool)] :
          ( pp(aa(set(A),bool,member(A,X),semiring_1_Nats(A)))
         => ( ! [N4: nat] : pp(aa(A,bool,P2,aa(nat,A,semiring_1_of_nat(A),N4)))
           => pp(aa(A,bool,P2,X)) ) ) ) ).

% Nats_induct
tff(fact_7982_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),semiring_1_Nats(A)))
         => ~ ! [N4: nat] : X != aa(nat,A,semiring_1_of_nat(A),N4) ) ) ).

% Nats_cases
tff(fact_7983_image2pI,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: A,Y: B,F3: fun(A,C),G3: fun(B,D)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X),Y))
     => pp(aa(D,bool,aa(C,fun(D,bool),bNF_Greatest_image2p(A,C,B,D,F3,G3,R2),aa(A,C,F3,X)),aa(B,D,G3,Y))) ) ).

% image2pI
tff(fact_7984_image2pE,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(A,B),G3: fun(C,D),R2: fun(A,fun(C,bool)),Fx: B,Gy: D] :
      ( pp(aa(D,bool,aa(B,fun(D,bool),bNF_Greatest_image2p(A,B,C,D,F3,G3,R2),Fx),Gy))
     => ~ ! [X3: A] :
            ( ( Fx = aa(A,B,F3,X3) )
           => ! [Y3: C] :
                ( ( Gy = aa(C,D,G3,Y3) )
               => ~ pp(aa(C,bool,aa(A,fun(C,bool),R2,X3),Y3)) ) ) ) ).

% image2pE
tff(fact_7985_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,member(A,one_one(A)),semiring_1_Nats(A))) ) ).

% Nats_1
tff(fact_7986_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_mult
tff(fact_7987_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,member(A,zero_zero(A)),semiring_1_Nats(A))) ) ).

% Nats_0
tff(fact_7988_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_add
tff(fact_7989_image2p__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(C,A),G3: fun(D,B),R2: fun(C,fun(D,bool)),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bNF_Greatest_image2p(C,A,D,B,F3,G3,R2),X5),Xa))
    <=> ? [X15: C,Y11: D] :
          ( pp(aa(D,bool,aa(C,fun(D,bool),R2,X15),Y11))
          & ( aa(C,A,F3,X15) = X5 )
          & ( aa(D,B,G3,Y11) = Xa ) ) ) ).

% image2p_def
tff(fact_7990_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_7991_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aua(A,bool)) ) ) ).

% Nats_altdef2
tff(fact_7992_subset__mset_Osup__Inf2__distrib,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B5))
         => ( ( B5 != bot_bot(set(multiset(A))) )
           => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A6)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B5)) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_aub(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),A6),B5))) ) ) ) ) ) ).

% subset_mset.sup_Inf2_distrib
tff(fact_7993_subset__mset_Osup__Inf1__distrib,axiom,
    ! [A: $tType,A6: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),X),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A6)) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_auc(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),A6),X))) ) ) ) ).

% subset_mset.sup_Inf1_distrib
tff(fact_7994_semilattice__inf_OInf__fin_Ocong,axiom,
    ! [A: $tType,Inf2: fun(A,fun(A,A))] : lattic8678736583308907530nf_fin(A,Inf2) = lattic8678736583308907530nf_fin(A,Inf2) ).

% semilattice_inf.Inf_fin.cong
tff(fact_7995_subset__mset_OInf__fin_Oeq__fold_H,axiom,
    ! [A: $tType,A6: set(multiset(A))] : aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A6) = aa(option(multiset(A)),multiset(A),the2(multiset(A)),finite_fold(multiset(A),option(multiset(A)),aTP_Lamp_aud(multiset(A),fun(option(multiset(A)),option(multiset(A)))),none(multiset(A)),A6)) ).

% subset_mset.Inf_fin.eq_fold'
tff(fact_7996_subset__mset_OInf__fin_Osubset,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( B5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),B5),A6))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B5)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A6)) = aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A6) ) ) ) ) ).

% subset_mset.Inf_fin.subset
tff(fact_7997_subset__mset_OInf__fin_Ounion,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B5))
         => ( ( B5 != bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A6)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B5)) ) ) ) ) ) ).

% subset_mset.Inf_fin.union
tff(fact_7998_subset__mset_OInf__fin_Osubset__imp,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A6),B5))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),B5)),aa(set(multiset(A)),multiset(A),lattic8678736583308907530nf_fin(multiset(A),inter_mset(A)),A6))) ) ) ) ).

% subset_mset.Inf_fin.subset_imp
tff(fact_7999_subset__mset_Oinf__Sup1__distrib,axiom,
    ! [A: $tType,A6: set(multiset(A)),X: multiset(A)] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),X),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A6)) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_aue(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),A6),X))) ) ) ) ).

% subset_mset.inf_Sup1_distrib
tff(fact_8000_subset__mset_Oinf__Sup2__distrib,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B5))
         => ( ( B5 != bot_bot(set(multiset(A))) )
           => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A6)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B5)) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_auf(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),A6),B5))) ) ) ) ) ) ).

% subset_mset.inf_Sup2_distrib
tff(fact_8001_subset__mset_OSup__fin_Oeq__fold_H,axiom,
    ! [A: $tType,A6: set(multiset(A))] : aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A6) = aa(option(multiset(A)),multiset(A),the2(multiset(A)),finite_fold(multiset(A),option(multiset(A)),aTP_Lamp_aug(multiset(A),fun(option(multiset(A)),option(multiset(A)))),none(multiset(A)),A6)) ).

% subset_mset.Sup_fin.eq_fold'
tff(fact_8002_semilattice__sup_OSup__fin_Ocong,axiom,
    ! [A: $tType,Sup2: fun(A,fun(A,A))] : lattic4630905495605216202up_fin(A,Sup2) = lattic4630905495605216202up_fin(A,Sup2) ).

% semilattice_sup.Sup_fin.cong
tff(fact_8003_subset__mset_OSup__fin_Osubset,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( B5 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),B5),A6))
         => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B5)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A6)) = aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A6) ) ) ) ) ).

% subset_mset.Sup_fin.subset
tff(fact_8004_subset__mset_OSup__fin_Ounion,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),A6))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B5))
         => ( ( B5 != bot_bot(set(multiset(A))) )
           => ( aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),aa(set(multiset(A)),set(multiset(A)),aa(set(multiset(A)),fun(set(multiset(A)),set(multiset(A))),sup_sup(set(multiset(A))),A6),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A6)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B5)) ) ) ) ) ) ).

% subset_mset.Sup_fin.union
tff(fact_8005_subset__mset_OSup__fin_Osubset__imp,axiom,
    ! [A: $tType,A6: set(multiset(A)),B5: set(multiset(A))] :
      ( pp(aa(set(multiset(A)),bool,aa(set(multiset(A)),fun(set(multiset(A)),bool),ord_less_eq(set(multiset(A))),A6),B5))
     => ( ( A6 != bot_bot(set(multiset(A))) )
       => ( pp(aa(set(multiset(A)),bool,finite_finite2(multiset(A)),B5))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),A6)),aa(set(multiset(A)),multiset(A),lattic4630905495605216202up_fin(multiset(A),union_mset(A)),B5))) ) ) ) ).

% subset_mset.Sup_fin.subset_imp
tff(fact_8006_flip__pred,axiom,
    ! [A: $tType,B: $tType,A6: set(product_prod(A,B)),R2: fun(B,fun(A,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),conversep(B,A,R2)))))
     => pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),aa(set(product_prod(A,B)),set(product_prod(B,A)),image2(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_pc(A,fun(B,product_prod(B,A))))),A6)),aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),R2)))) ) ).

% flip_pred
tff(fact_8007_finite__subset__Union__chain,axiom,
    ! [A: $tType,A6: set(A),B12: set(set(A)),A18: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)))
       => ( ( B12 != bot_bot(set(set(A))) )
         => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12))
           => ~ ! [B8: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),B8),B12))
                 => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B8)) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_8008_conversep__noteq,axiom,
    ! [A: $tType,X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),conversep(A,A,aTP_Lamp_amf(A,fun(A,bool))),X5),Xa))
    <=> ( X5 != Xa ) ) ).

% conversep_noteq
tff(fact_8009_conversep__mono,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,fun(A,bool)),S2: fun(B,fun(A,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),conversep(B,A,R3)),conversep(B,A,S2)))
    <=> pp(aa(fun(B,fun(A,bool)),bool,aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),bool),ord_less_eq(fun(B,fun(A,bool))),R3),S2)) ) ).

% conversep_mono
tff(fact_8010_converse__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : converse(A,B,X5) = aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)))) ).

% converse_def
tff(fact_8011_subset_Ochain__total,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A)),X: set(A),Y: set(A)] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
     => ( pp(aa(set(set(A)),bool,member(set(A),X),C4))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y),C4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X),Y))
            | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y),X)) ) ) ) ) ).

% subset.chain_total
tff(fact_8012_chains__alt__def,axiom,
    ! [A: $tType,A6: set(set(A))] : chains2(A,A6) = aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),A6,ord_less(set(A)))) ).

% chains_alt_def
tff(fact_8013_converse__join,axiom,
    ! [A: $tType,B: $tType,R3: fun(B,fun(A,bool)),S2: fun(B,fun(A,bool))] : conversep(B,A,aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),sup_sup(fun(B,fun(A,bool))),R3),S2)) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),conversep(B,A,R3)),conversep(B,A,S2)) ).

% converse_join
tff(fact_8014_pred__on_Ochain__total,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,fun(A,bool)),C4: set(A),X: A,Y: A] :
      ( pp(aa(set(A),bool,pred_chain(A,A6,P2),C4))
     => ( pp(aa(set(A),bool,member(A,X),C4))
       => ( pp(aa(set(A),bool,member(A,Y),C4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)),X),Y))
            | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)),Y),X)) ) ) ) ) ).

% pred_on.chain_total
tff(fact_8015_subset_Ochain__empty,axiom,
    ! [A: $tType,A6: set(set(A))] : pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),bot_bot(set(set(A))))) ).

% subset.chain_empty
tff(fact_8016_conversep__converse__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),X5: B,Xa: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),X5),Xa))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X5),Xa)),converse(A,B,R3))) ) ).

% conversep_converse_eq
tff(fact_8017_subset__chain__def,axiom,
    ! [A: $tType,A18: set(set(A)),C9: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),C9))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C9),A18))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),C9))
           => ! [Xa3: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),Xa3),C9))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),Xa3))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa3),X4)) ) ) ) ) ) ).

% subset_chain_def
tff(fact_8018_subset_Ochain__def,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C4),A6))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),C4))
           => ! [Xa3: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),Xa3),C4))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X4),Xa3))
                  | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Xa3),X4)) ) ) ) ) ) ).

% subset.chain_def
tff(fact_8019_subset_OchainI,axiom,
    ! [A: $tType,C4: set(set(A)),A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C4),A6))
     => ( ! [X3: set(A),Y3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),C4))
           => ( pp(aa(set(set(A)),bool,member(set(A),Y3),C4))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X3),Y3))
                | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y3),X3)) ) ) )
       => pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4)) ) ) ).

% subset.chainI
tff(fact_8020_subset__Zorn,axiom,
    ! [A: $tType,A6: set(set(A))] :
      ( ! [C8: set(set(A))] :
          ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C8))
         => ? [X5: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X5),A6))
              & ! [Xa4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),Xa4),C8))
                 => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa4),X5)) ) ) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A6))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa),A6))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa))
               => ( Xa = X3 ) ) ) ) ) ).

% subset_Zorn
tff(fact_8021_subset__Zorn_H,axiom,
    ! [A: $tType,A6: set(set(A))] :
      ( ! [C8: set(set(A))] :
          ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C8))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8)),A6)) )
     => ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),A6))
          & ! [Xa: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),Xa),A6))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa))
               => ( Xa = X3 ) ) ) ) ) ).

% subset_Zorn'
tff(fact_8022_Quotient__composition__le__eq,axiom,
    ! [B: $tType,A: $tType,T8: fun(A,fun(B,bool)),R2: fun(B,fun(B,bool))] :
      ( left_unique(A,B,T8)
     => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),R2),fequal(B)))
       => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T8),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),R2),conversep(A,B,T8)))),fequal(A))) ) ) ).

% Quotient_composition_le_eq
tff(fact_8023_left__unique__alt__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( left_unique(A,B,R2)
    <=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R2),conversep(A,B,R2))),fequal(A))) ) ).

% left_unique_alt_def
tff(fact_8024_right__total__alt__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( right_total(A,B,R2)
    <=> pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),fequal(B)),aa(fun(A,fun(B,bool)),fun(B,fun(B,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(B,fun(B,bool))),relcompp(B,A,B),conversep(A,B,R2)),R2))) ) ).

% right_total_alt_def
tff(fact_8025_conversep__le__swap,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),S2: fun(B,fun(A,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R3),conversep(B,A,S2)))
    <=> pp(aa(fun(B,fun(A,bool)),bool,aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),bool),ord_less_eq(fun(B,fun(A,bool))),conversep(A,B,R3)),S2)) ) ).

% conversep_le_swap
tff(fact_8026_leq__conversepI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( ( R2 = fequal(A) )
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R2),conversep(A,A,R2))) ) ).

% leq_conversepI
tff(fact_8027_pred__on_Ochain__def,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,fun(A,bool)),C4: set(A)] :
      ( pp(aa(set(A),bool,pred_chain(A,A6,P2),C4))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A6))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),C4))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),C4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)),X4),Xa3))
                  | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)),Xa3),X4)) ) ) ) ) ) ).

% pred_on.chain_def
tff(fact_8028_pred__on_OchainI,axiom,
    ! [A: $tType,C4: set(A),A6: set(A),P2: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A6))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(A),bool,member(A,X3),C4))
           => ( pp(aa(set(A),bool,member(A,Y3),C4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)),X3),Y3))
                | pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)),Y3),X3)) ) ) )
       => pp(aa(set(A),bool,pred_chain(A,A6,P2),C4)) ) ) ).

% pred_on.chainI
tff(fact_8029_left__total__alt__def,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( left_total(A,B,R2)
    <=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R2),conversep(A,B,R2)))) ) ).

% left_total_alt_def
tff(fact_8030_Quotient__composition__ge__eq,axiom,
    ! [B: $tType,A: $tType,T8: fun(A,fun(B,bool)),R2: fun(B,fun(B,bool))] :
      ( left_total(A,B,T8)
     => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),fequal(B)),R2))
       => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T8),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),R2),conversep(A,B,T8))))) ) ) ).

% Quotient_composition_ge_eq
tff(fact_8031_right__unique__alt__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] :
      ( right_unique(A,B,R2)
    <=> pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),aa(fun(A,fun(B,bool)),fun(B,fun(B,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(B,fun(B,bool))),relcompp(B,A,B),conversep(A,B,R2)),R2)),fequal(B))) ) ).

% right_unique_alt_def
tff(fact_8032_subset__chain__insert,axiom,
    ! [A: $tType,A18: set(set(A)),B5: set(A),B12: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),B5),B12)))
    <=> ( pp(aa(set(set(A)),bool,member(set(A),B5),A18))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),B12))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X4),B5))
              | pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),X4)) ) )
        & pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12)) ) ) ).

% subset_chain_insert
tff(fact_8033_chain__subset__alt__def,axiom,
    ! [A: $tType,C4: set(set(A))] :
      ( chain_subset(A,C4)
    <=> pp(aa(set(set(A)),bool,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C4)) ) ).

% chain_subset_alt_def
tff(fact_8034_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A18: set(set(A))] :
      ( ( A18 != bot_bot(set(set(A))) )
     => ( ! [C10: set(set(A))] :
            ( ( C10 != bot_bot(set(set(A))) )
           => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),C10))
             => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C10)),A18)) ) )
       => ? [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X3),A18))
            & ! [Xa: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),Xa),A18))
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Xa))
                 => ( Xa = X3 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_8035_Union__in__chain,axiom,
    ! [A: $tType,B12: set(set(A)),A18: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B12))
     => ( ( B12 != bot_bot(set(set(A))) )
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)),B12)) ) ) ) ).

% Union_in_chain
tff(fact_8036_Inter__in__chain,axiom,
    ! [A: $tType,B12: set(set(A)),A18: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B12))
     => ( ( B12 != bot_bot(set(set(A))) )
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A18,ord_less(set(A))),B12))
         => pp(aa(set(set(A)),bool,member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)),B12)) ) ) ) ).

% Inter_in_chain
tff(fact_8037_subset_Ochain__extend,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A)),Z4: set(A)] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
     => ( pp(aa(set(set(A)),bool,member(set(A),Z4),A6))
       => ( ! [X3: set(A)] :
              ( pp(aa(set(set(A)),bool,member(set(A),X3),C4))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X3),Z4)) )
         => pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(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))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Z4),bot_bot(set(set(A))))),C4))) ) ) ) ).

% subset.chain_extend
tff(fact_8038_pred__on_Ochain__extend,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,fun(A,bool)),C4: set(A),Z4: A] :
      ( pp(aa(set(A),bool,pred_chain(A,A6,P2),C4))
     => ( pp(aa(set(A),bool,member(A,Z4),A6))
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),C4))
             => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),P2),fequal(A)),X3),Z4)) )
         => pp(aa(set(A),bool,pred_chain(A,A6,P2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z4),bot_bot(set(A)))),C4))) ) ) ) ).

% pred_on.chain_extend
tff(fact_8039_Chains__subset_H,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R3)
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)))),chains(A,R3))) ) ).

% Chains_subset'
tff(fact_8040_multiset_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : rel_mset(A,B,R2) = aa(fun(multiset(product_prod(A,B)),fun(multiset(B),bool)),fun(multiset(A),fun(multiset(B),bool)),aa(fun(multiset(A),fun(multiset(product_prod(A,B)),bool)),fun(fun(multiset(product_prod(A,B)),fun(multiset(B),bool)),fun(multiset(A),fun(multiset(B),bool))),relcompp(multiset(A),multiset(product_prod(A,B)),multiset(B)),conversep(multiset(product_prod(A,B)),multiset(A),bNF_Grp(multiset(product_prod(A,B)),multiset(A),aa(fun(multiset(product_prod(A,B)),bool),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_akg(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),R2)),image_mset(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(multiset(product_prod(A,B)),multiset(B),aa(fun(multiset(product_prod(A,B)),bool),set(multiset(product_prod(A,B))),collect(multiset(product_prod(A,B))),aTP_Lamp_akg(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),R2)),image_mset(product_prod(A,B),B,product_snd(A,B)))) ).

% multiset.rel_compp_Grp
tff(fact_8041_OO__Grp__cong,axiom,
    ! [B: $tType,C: $tType,A: $tType,A6: set(A),B5: set(A),F3: fun(A,B),G3: fun(A,C)] :
      ( ( A6 = B5 )
     => ( aa(fun(A,fun(C,bool)),fun(B,fun(C,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,bool))),relcompp(B,A,C),conversep(A,B,bNF_Grp(A,B,A6,F3))),bNF_Grp(A,C,A6,G3)) = aa(fun(A,fun(C,bool)),fun(B,fun(C,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,bool))),relcompp(B,A,C),conversep(A,B,bNF_Grp(A,B,B5,F3))),bNF_Grp(A,C,B5,G3)) ) ) ).

% OO_Grp_cong
tff(fact_8042_eq__le__Grp__id__iff,axiom,
    ! [A: $tType,R2: fun(A,bool)] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),bNF_Grp(A,A,aa(fun(A,bool),set(A),collect(A),R2),id(A))))
    <=> ! [X_12: A] : pp(aa(A,bool,R2,X_12)) ) ).

% eq_le_Grp_id_iff
tff(fact_8043_fun_Orel__Grp,axiom,
    ! [D: $tType,B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] : bNF_rel_fun(D,D,A,B,fequal(D),bNF_Grp(A,B,A6,F3)) = bNF_Grp(fun(D,A),fun(D,B),aa(fun(fun(D,A),bool),set(fun(D,A)),collect(fun(D,A)),aTP_Lamp_auh(set(A),fun(fun(D,A),bool),A6)),comp(A,B,D,F3)) ).

% fun.rel_Grp
tff(fact_8044_list_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] : list_all2(A,B,bNF_Grp(A,B,A6,F3)) = bNF_Grp(list(A),list(B),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_aoz(set(A),fun(list(A),bool),A6)),map(A,B,F3)) ).

% list.rel_Grp
tff(fact_8045_Grp__mono,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B5: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B5))
     => pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),bNF_Grp(A,B,A6,F3)),bNF_Grp(A,B,B5,F3))) ) ).

% Grp_mono
tff(fact_8046_multiset_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] : rel_mset(A,B,bNF_Grp(A,B,A6,F3)) = bNF_Grp(multiset(A),multiset(B),aa(fun(multiset(A),bool),set(multiset(A)),collect(multiset(A)),aTP_Lamp_aui(set(A),fun(multiset(A),bool),A6)),image_mset(A,B,F3)) ).

% multiset.rel_Grp
tff(fact_8047_Grp__def,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bNF_Grp(A,B,A6,F3),X5),Xa))
    <=> ( ( Xa = aa(A,B,F3,X5) )
        & pp(aa(set(A),bool,member(A,X5),A6)) ) ) ).

% Grp_def
tff(fact_8048_OO__Grp__alt,axiom,
    ! [A: $tType,C: $tType,B: $tType,A6: set(C),F3: fun(C,A),G3: fun(C,B),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),conversep(C,A,bNF_Grp(C,A,A6,F3))),bNF_Grp(C,B,A6,G3)),X5),Xa))
    <=> ? [Z2: C] :
          ( pp(aa(set(C),bool,member(C,Z2),A6))
          & ( aa(C,A,F3,Z2) = X5 )
          & ( aa(C,B,G3,Z2) = Xa ) ) ) ).

% OO_Grp_alt
tff(fact_8049_mono__Chains,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S2))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R3)),chains(A,S2))) ) ).

% mono_Chains
tff(fact_8050_Chains__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : chains(A,R3) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_auj(set(product_prod(A,A)),fun(set(A),bool),R3)) ).

% Chains_def
tff(fact_8051_Chains__relation__of,axiom,
    ! [A: $tType,C4: set(A),P2: fun(A,fun(A,bool)),A6: set(A)] :
      ( pp(aa(set(set(A)),bool,member(set(A),C4),chains(A,order_relation_of(A,P2,A6))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C4),A6)) ) ).

% Chains_relation_of
tff(fact_8052_Chains__inits__DiffI,axiom,
    ! [A: $tType,R2: set(set(product_prod(A,A))),S2: set(product_prod(A,A))] :
      ( pp(aa(set(set(set(product_prod(A,A)))),bool,member(set(set(product_prod(A,A))),R2),chains(set(product_prod(A,A)),init_seg_of(A))))
     => pp(aa(set(set(set(product_prod(A,A)))),bool,member(set(set(product_prod(A,A))),aa(fun(set(product_prod(A,A)),bool),set(set(product_prod(A,A))),collect(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_auk(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),R2),S2))),chains(set(product_prod(A,A)),init_seg_of(A)))) ) ).

% Chains_inits_DiffI
tff(fact_8053_Chains__alt__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R3)
     => ( chains(A,R3) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))) ) ) ).

% Chains_alt_def
tff(fact_8054_Chains__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R3)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))))) ).

% Chains_subset
tff(fact_8055_fun_Orel__compp__Grp,axiom,
    ! [D: $tType,B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : bNF_rel_fun(D,D,A,B,fequal(D),R2) = aa(fun(fun(D,product_prod(A,B)),fun(fun(D,B),bool)),fun(fun(D,A),fun(fun(D,B),bool)),aa(fun(fun(D,A),fun(fun(D,product_prod(A,B)),bool)),fun(fun(fun(D,product_prod(A,B)),fun(fun(D,B),bool)),fun(fun(D,A),fun(fun(D,B),bool))),relcompp(fun(D,A),fun(D,product_prod(A,B)),fun(D,B)),conversep(fun(D,product_prod(A,B)),fun(D,A),bNF_Grp(fun(D,product_prod(A,B)),fun(D,A),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_aaw(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R2)),comp(product_prod(A,B),A,D,product_fst(A,B))))),bNF_Grp(fun(D,product_prod(A,B)),fun(D,B),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_aaw(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R2)),comp(product_prod(A,B),B,D,product_snd(A,B)))) ).

% fun.rel_compp_Grp
tff(fact_8056_list_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : list_all2(A,B,R2) = aa(fun(list(product_prod(A,B)),fun(list(B),bool)),fun(list(A),fun(list(B),bool)),aa(fun(list(A),fun(list(product_prod(A,B)),bool)),fun(fun(list(product_prod(A,B)),fun(list(B),bool)),fun(list(A),fun(list(B),bool))),relcompp(list(A),list(product_prod(A,B)),list(B)),conversep(list(product_prod(A,B)),list(A),bNF_Grp(list(product_prod(A,B)),list(A),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ajl(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R2)),map(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(list(product_prod(A,B)),list(B),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ajl(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R2)),map(product_prod(A,B),B,product_snd(A,B)))) ).

% list.rel_compp_Grp
tff(fact_8057_Grp__fst__snd,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : aa(fun(product_prod(A,B),fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(product_prod(A,B),bool)),fun(fun(product_prod(A,B),fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,product_prod(A,B),B),conversep(product_prod(A,B),A,bNF_Grp(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_fst(A,B)))),bNF_Grp(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)),product_snd(A,B))) = R2 ).

% Grp_fst_snd
tff(fact_8058_pred__on_Onot__maxchain__Some,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,fun(A,bool)),C4: set(A)] :
      ( pp(aa(set(A),bool,pred_chain(A,A6,P2),C4))
     => ( ~ pred_maxchain(A,A6,P2,C4)
       => ( pp(aa(set(A),bool,pred_chain(A,A6,P2),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_aul(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A6),P2),C4))))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),C4),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_aul(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A6),P2),C4)))) ) ) ) ).

% pred_on.not_maxchain_Some
tff(fact_8059_type__copy__vimage2p__Grp__Abs,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: fun(A,B),Abs: fun(B,A),G3: fun(D,C),P2: fun(C,bool),H: fun(C,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( bNF_vimage2p(D,C,B,A,bool,G3,Abs,bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),P2),H)) = bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),aa(fun(C,bool),fun(D,bool),aTP_Lamp_aum(fun(D,C),fun(fun(C,bool),fun(D,bool)),G3),P2)),aa(fun(D,C),fun(D,B),comp(C,B,D,aa(fun(C,A),fun(C,B),comp(A,B,C,Rep),H)),G3)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_8060_vimage2p__relcompp__mono,axiom,
    ! [C: $tType,F2: $tType,E: $tType,D: $tType,B: $tType,A: $tType,R2: fun(A,fun(C,bool)),S: fun(C,fun(B,bool)),T8: fun(A,fun(B,bool)),F3: fun(D,A),G3: fun(F2,C),H: fun(E,B)] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R2),S)),T8))
     => pp(aa(fun(D,fun(E,bool)),bool,aa(fun(D,fun(E,bool)),fun(fun(D,fun(E,bool)),bool),ord_less_eq(fun(D,fun(E,bool))),aa(fun(F2,fun(E,bool)),fun(D,fun(E,bool)),aa(fun(D,fun(F2,bool)),fun(fun(F2,fun(E,bool)),fun(D,fun(E,bool))),relcompp(D,F2,E),bNF_vimage2p(D,A,F2,C,bool,F3,G3,R2)),bNF_vimage2p(F2,C,E,B,bool,G3,H,S))),bNF_vimage2p(D,A,E,B,bool,F3,H,T8))) ) ).

% vimage2p_relcompp_mono
tff(fact_8061_rel__fun__iff__leq__vimage2p,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,R2: fun(A,fun(B,bool)),S: fun(C,fun(D,bool)),F3: fun(A,C),G3: fun(B,D)] :
      ( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,R2,S),F3),G3))
    <=> pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),bNF_vimage2p(A,C,B,D,bool,F3,G3,S))) ) ).

% rel_fun_iff_leq_vimage2p
tff(fact_8062_vimage2p__mono,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,F3: fun(A,B),G3: fun(C,D),R2: fun(B,fun(D,bool)),X: A,Y: C,S: fun(B,fun(D,bool))] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),bNF_vimage2p(A,B,C,D,bool,F3,G3,R2),X),Y))
     => ( pp(aa(fun(B,fun(D,bool)),bool,aa(fun(B,fun(D,bool)),fun(fun(B,fun(D,bool)),bool),ord_less_eq(fun(B,fun(D,bool))),R2),S))
       => pp(aa(C,bool,aa(A,fun(C,bool),bNF_vimage2p(A,B,C,D,bool,F3,G3,S),X),Y)) ) ) ).

% vimage2p_mono
tff(fact_8063_predicate2D__vimage2p,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R2: fun(A,fun(B,bool)),F3: fun(A,C),G3: fun(B,D),S: fun(C,fun(D,bool)),X: A,Y: B] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),bNF_vimage2p(A,C,B,D,bool,F3,G3,S)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X),Y))
       => pp(aa(D,bool,aa(C,fun(D,bool),S,aa(A,C,F3,X)),aa(B,D,G3,Y))) ) ) ).

% predicate2D_vimage2p
tff(fact_8064_vimage2p__def,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F3: fun(A,D),G3: fun(B,E),R2: fun(D,fun(E,C)),X5: A,Xa: B] : aa(B,C,aa(A,fun(B,C),bNF_vimage2p(A,D,B,E,C,F3,G3,R2),X5),Xa) = aa(E,C,aa(D,fun(E,C),R2,aa(A,D,F3,X5)),aa(B,E,G3,Xa)) ).

% vimage2p_def
tff(fact_8065_vimage2p__cong,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,R2: fun(A,fun(B,C)),S: fun(A,fun(B,C)),F3: fun(D,A),G3: fun(E,B)] :
      ( ( R2 = S )
     => ( bNF_vimage2p(D,A,E,B,C,F3,G3,R2) = bNF_vimage2p(D,A,E,B,C,F3,G3,S) ) ) ).

% vimage2p_cong
tff(fact_8066_subset_OHausdorff,axiom,
    ! [A: $tType,A6: set(set(A))] :
    ? [X_1: set(set(A))] : pred_maxchain(set(A),A6,ord_less(set(A)),X_1) ).

% subset.Hausdorff
tff(fact_8067_subset_Omaxchain__def,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A))] :
      ( pred_maxchain(set(A),A6,ord_less(set(A)),C4)
    <=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
        & ~ ? [S9: set(set(A))] :
              ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),S9))
              & pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),C4),S9)) ) ) ) ).

% subset.maxchain_def
tff(fact_8068_subset_Omaxchain__imp__chain,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A))] :
      ( pred_maxchain(set(A),A6,ord_less(set(A)),C4)
     => pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4)) ) ).

% subset.maxchain_imp_chain
tff(fact_8069_subset_Onot__maxchain__Some,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
     => ( ~ pred_maxchain(set(A),A6,ord_less(set(A)),C4)
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aun(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A6),C4))))
          & pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),C4),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aun(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A6),C4)))) ) ) ) ).

% subset.not_maxchain_Some
tff(fact_8070_pred__on_Omaxchain__def,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,fun(A,bool)),C4: set(A)] :
      ( pred_maxchain(A,A6,P2,C4)
    <=> ( pp(aa(set(A),bool,pred_chain(A,A6,P2),C4))
        & ~ ? [S9: set(A)] :
              ( pp(aa(set(A),bool,pred_chain(A,A6,P2),S9))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),C4),S9)) ) ) ) ).

% pred_on.maxchain_def
tff(fact_8071_subset__maxchain__max,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A)),X6: set(A)] :
      ( pred_maxchain(set(A),A6,ord_less(set(A)),C4)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),A6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)),X6))
         => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4) = X6 ) ) ) ) ).

% subset_maxchain_max
tff(fact_8072_vimage2p__relcompp__converse,axiom,
    ! [E: $tType,C: $tType,D: $tType,A: $tType,F2: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),F3: fun(C,E),G3: fun(D,F2),R2: fun(B,fun(E,bool)),S: fun(B,fun(F2,bool))] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( bNF_vimage2p(C,E,D,F2,bool,F3,G3,aa(fun(B,fun(F2,bool)),fun(E,fun(F2,bool)),aa(fun(E,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(E,fun(F2,bool))),relcompp(E,B,F2),conversep(B,E,R2)),S)) = aa(fun(A,fun(D,bool)),fun(C,fun(D,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(D,bool)),fun(C,fun(D,bool))),relcompp(C,A,D),conversep(A,C,bNF_vimage2p(A,B,C,E,bool,Rep,F3,R2))),bNF_vimage2p(A,B,D,F2,bool,Rep,G3,S)) ) ) ).

% vimage2p_relcompp_converse
tff(fact_8073_type__copy__vimage2p__Grp__Rep,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),F3: fun(C,D),P2: fun(D,bool),H: fun(D,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( bNF_vimage2p(C,D,A,B,bool,F3,Rep,bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),P2),H)) = bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),aa(fun(D,bool),fun(C,bool),aTP_Lamp_auo(fun(C,D),fun(fun(D,bool),fun(C,bool)),F3),P2)),aa(fun(C,D),fun(C,A),comp(D,A,C,aa(fun(D,B),fun(D,A),comp(B,A,D,Abs),H)),F3)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_8074_convol__image__vimage2p,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,F3: fun(C,A),G3: fun(D,B),R2: fun(A,fun(B,bool))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),bNF_convol(product_prod(C,D),A,B,aa(fun(product_prod(C,D),C),fun(product_prod(C,D),A),comp(C,A,product_prod(C,D),F3),product_fst(C,D)),aa(fun(product_prod(C,D),D),fun(product_prod(C,D),B),comp(D,B,product_prod(C,D),G3),product_snd(C,D)))),aa(fun(product_prod(C,D),bool),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),bNF_vimage2p(C,A,D,B,bool,F3,G3,R2))))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R2)))) ).

% convol_image_vimage2p
tff(fact_8075_pred__on_Osuc__def,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,fun(A,bool)),C4: set(A)] :
      ( ( ( ~ pp(aa(set(A),bool,pred_chain(A,A6,P2),C4))
          | pred_maxchain(A,A6,P2,C4) )
       => ( pred_suc(A,A6,P2,C4) = C4 ) )
      & ( ~ ( ~ pp(aa(set(A),bool,pred_chain(A,A6,P2),C4))
            | pred_maxchain(A,A6,P2,C4) )
       => ( pred_suc(A,A6,P2,C4) = fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_aul(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A6),P2),C4)) ) ) ) ).

% pred_on.suc_def
tff(fact_8076_subset_Onot__chain__suc,axiom,
    ! [A: $tType,A6: set(set(A)),X6: set(set(A))] :
      ( ~ pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),X6))
     => ( pred_suc(set(A),A6,ord_less(set(A)),X6) = X6 ) ) ).

% subset.not_chain_suc
tff(fact_8077_subset_Omaxchain__suc,axiom,
    ! [A: $tType,A6: set(set(A)),X6: set(set(A))] :
      ( pred_maxchain(set(A),A6,ord_less(set(A)),X6)
     => ( pred_suc(set(A),A6,ord_less(set(A)),X6) = X6 ) ) ).

% subset.maxchain_suc
tff(fact_8078_subset_Osuc__not__equals,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
     => ( ~ pred_maxchain(set(A),A6,ord_less(set(A)),C4)
       => ( pred_suc(set(A),A6,ord_less(set(A)),C4) != C4 ) ) ) ).

% subset.suc_not_equals
tff(fact_8079_subset_Osuc__def,axiom,
    ! [A: $tType,A6: set(set(A)),C4: set(set(A))] :
      ( ( ( ~ pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
          | pred_maxchain(set(A),A6,ord_less(set(A)),C4) )
       => ( pred_suc(set(A),A6,ord_less(set(A)),C4) = C4 ) )
      & ( ~ ( ~ pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),C4))
            | pred_maxchain(set(A),A6,ord_less(set(A)),C4) )
       => ( pred_suc(set(A),A6,ord_less(set(A)),C4) = fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aun(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A6),C4)) ) ) ) ).

% subset.suc_def
tff(fact_8080_subset_Ochain__suc,axiom,
    ! [A: $tType,A6: set(set(A)),X6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),X6))
     => pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),pred_suc(set(A),A6,ord_less(set(A)),X6))) ) ).

% subset.chain_suc
tff(fact_8081_pred__on_Ochain__sucD,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,fun(A,bool)),X6: set(A)] :
      ( pp(aa(set(A),bool,pred_chain(A,A6,P2),X6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),pred_suc(A,A6,P2,X6)),A6))
        & pp(aa(set(A),bool,pred_chain(A,A6,P2),pred_suc(A,A6,P2,X6))) ) ) ).

% pred_on.chain_sucD
tff(fact_8082_convol__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,B),G3: fun(A,C),X5: A] : aa(A,product_prod(B,C),bNF_convol(A,B,C,F3,G3),X5) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,F3,X5)),aa(A,C,G3,X5)) ).

% convol_def
tff(fact_8083_subset_Osubset__suc,axiom,
    ! [A: $tType,X6: set(set(A)),Y6: set(set(A)),A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),Y6))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),pred_suc(set(A),A6,ord_less(set(A)),Y6))) ) ).

% subset.subset_suc
tff(fact_8084_subset_Osuc__subset,axiom,
    ! [A: $tType,X6: set(set(A)),A6: set(set(A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),pred_suc(set(A),A6,ord_less(set(A)),X6))) ).

% subset.suc_subset
tff(fact_8085_subset_Osuc__in__carrier,axiom,
    ! [A: $tType,X6: set(set(A)),A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),A6))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pred_suc(set(A),A6,ord_less(set(A)),X6)),A6)) ) ).

% subset.suc_in_carrier
tff(fact_8086_pred__on_Osubset__suc,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),A6: set(A),P2: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),Y6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),pred_suc(A,A6,P2,Y6))) ) ).

% pred_on.subset_suc
tff(fact_8087_pred__on_Osuc__subset,axiom,
    ! [A: $tType,X6: set(A),A6: set(A),P2: fun(A,fun(A,bool))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),pred_suc(A,A6,P2,X6))) ).

% pred_on.suc_subset
tff(fact_8088_pred__on_Osuc__in__carrier,axiom,
    ! [A: $tType,X6: set(A),A6: set(A),P2: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A6))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),pred_suc(A,A6,P2,X6)),A6)) ) ).

% pred_on.suc_in_carrier
tff(fact_8089_subset_Ochain__sucD,axiom,
    ! [A: $tType,A6: set(set(A)),X6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),X6))
     => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pred_suc(set(A),A6,ord_less(set(A)),X6)),A6))
        & pp(aa(set(set(A)),bool,pred_chain(set(A),A6,ord_less(set(A))),pred_suc(set(A),A6,ord_less(set(A)),X6))) ) ) ).

% subset.chain_sucD
tff(fact_8090_Quotient__alt__def5,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T8: fun(A,fun(B,bool))] :
      ( quotient(A,B,R2,Abs,Rep,T8)
    <=> ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),T8),bNF_Grp(A,B,top_top(set(A)),Abs)))
        & pp(aa(fun(B,fun(A,bool)),bool,aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),bool),ord_less_eq(fun(B,fun(A,bool))),bNF_Grp(B,A,top_top(set(B)),Rep)),conversep(A,B,T8)))
        & ( R2 = aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T8),conversep(A,B,T8)) ) ) ) ).

% Quotient_alt_def5
tff(fact_8091_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( transp(A,P2)
     => ( sorted_wrt(A,P2,Xs)
      <=> ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I)))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_8092_transp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less(A)) ) ).

% transp_less
tff(fact_8093_transp__gr,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_ax(A,fun(A,bool))) ) ).

% transp_gr
tff(fact_8094_Quotient__rep__abs__eq,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T8: fun(A,fun(B,bool)),T5: A] :
      ( quotient(A,B,R2,Abs,Rep,T8)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),R2,T5),T5))
       => ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R2),fequal(A)))
         => ( aa(B,A,Rep,aa(A,B,Abs,T5)) = T5 ) ) ) ) ).

% Quotient_rep_abs_eq
tff(fact_8095_transp__ge,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_aup(A,fun(A,bool))) ) ).

% transp_ge
tff(fact_8096_transp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less_eq(A)) ) ).

% transp_le
tff(fact_8097_transp__trans__eq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( transp(A,aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))
    <=> trans(A,R3) ) ).

% transp_trans_eq
tff(fact_8098_UNIV__typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),T8: fun(B,fun(A,bool))] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ! [X3: B,Xa4: A] :
            ( pp(aa(A,bool,aa(B,fun(A,bool),T8,X3),Xa4))
          <=> ( X3 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,fequal(B),Abs,Rep,T8) ) ) ).

% UNIV_typedef_to_Quotient
tff(fact_8099_subset__mset_Otransp__ge,axiom,
    ! [A: $tType] : transp(multiset(A),aTP_Lamp_atv(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.transp_ge
tff(fact_8100_transp__singleton,axiom,
    ! [A: $tType,A3: A] : transp(A,aTP_Lamp_auq(A,fun(A,fun(A,bool)),A3)) ).

% transp_singleton
tff(fact_8101_transp__empty,axiom,
    ! [A: $tType] : transp(A,aTP_Lamp_aur(A,fun(A,bool))) ).

% transp_empty
tff(fact_8102_Quotient__cr__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T8: fun(A,fun(B,bool))] :
      ( quotient(A,B,R2,Abs,Rep,T8)
     => ! [X5: A,Xa: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X5),Xa))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X5),X5))
            & ( aa(A,B,Abs,X5) = Xa ) ) ) ) ).

% Quotient_cr_rel
tff(fact_8103_Quotient__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T8: fun(A,fun(B,bool))] :
      ( quotient(A,B,R2,Abs,Rep,T8)
    <=> ( ! [A9: B] : aa(A,B,Abs,aa(B,A,Rep,A9)) = A9
        & ! [A9: B] : pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,Rep,A9)),aa(B,A,Rep,A9)))
        & ! [R4: A,S7: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),R2,R4),S7))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R2,R4),R4))
              & pp(aa(A,bool,aa(A,fun(A,bool),R2,S7),S7))
              & ( aa(A,B,Abs,R4) = aa(A,B,Abs,S7) ) ) )
        & ! [X4: A,Xa3: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X4),Xa3))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X4),X4))
              & ( aa(A,B,Abs,X4) = Xa3 ) ) ) ) ) ).

% Quotient_def
tff(fact_8104_QuotientI,axiom,
    ! [A: $tType,B: $tType,Abs: fun(B,A),Rep: fun(A,B),R2: fun(B,fun(B,bool)),T8: fun(B,fun(A,bool))] :
      ( ! [A5: A] : aa(B,A,Abs,aa(A,B,Rep,A5)) = A5
     => ( ! [A5: A] : pp(aa(B,bool,aa(B,fun(B,bool),R2,aa(A,B,Rep,A5)),aa(A,B,Rep,A5)))
       => ( ! [R: B,S5: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),R2,R),S5))
            <=> ( pp(aa(B,bool,aa(B,fun(B,bool),R2,R),R))
                & pp(aa(B,bool,aa(B,fun(B,bool),R2,S5),S5))
                & ( aa(B,A,Abs,R) = aa(B,A,Abs,S5) ) ) )
         => ( ! [X3: B,Xa4: A] :
                ( pp(aa(A,bool,aa(B,fun(A,bool),T8,X3),Xa4))
              <=> ( pp(aa(B,bool,aa(B,fun(B,bool),R2,X3),X3))
                  & ( aa(B,A,Abs,X3) = Xa4 ) ) )
           => quotient(B,A,R2,Abs,Rep,T8) ) ) ) ) ).

% QuotientI
tff(fact_8105_open__typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),P2: fun(B,bool),T8: fun(B,fun(A,bool))] :
      ( type_definition(A,B,Rep,Abs,aa(fun(B,bool),set(B),collect(B),P2))
     => ( ! [X3: B,Xa4: A] :
            ( pp(aa(A,bool,aa(B,fun(A,bool),T8,X3),Xa4))
          <=> ( X3 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,bNF_eq_onp(B,P2),Abs,Rep,T8) ) ) ).

% open_typedef_to_Quotient
tff(fact_8106_typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S: set(B),T8: fun(B,fun(A,bool))] :
      ( type_definition(A,B,Rep,Abs,S)
     => ( ! [X3: B,Xa4: A] :
            ( pp(aa(A,bool,aa(B,fun(A,bool),T8,X3),Xa4))
          <=> ( X3 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,bNF_eq_onp(B,aTP_Lamp_be(set(B),fun(B,bool),S)),Abs,Rep,T8) ) ) ).

% typedef_to_Quotient
tff(fact_8107_subset__mset_Otransp__gr,axiom,
    ! [A: $tType] : transp(multiset(A),aTP_Lamp_atw(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.transp_gr
tff(fact_8108_transp__trans,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] :
      ( transp(A,R3)
    <=> trans(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))) ) ).

% transp_trans
tff(fact_8109_transp__relcompp__less__eq,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] :
      ( transp(A,R3)
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R3),R3)),R3)) ) ).

% transp_relcompp_less_eq
tff(fact_8110_transp__relcompp,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] :
      ( transp(A,R3)
    <=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R3),R3)),R3)) ) ).

% transp_relcompp
tff(fact_8111_multp__implies__one__step,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),M6: multiset(A),N7: multiset(A)] :
      ( transp(A,R3)
     => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),M6),N7))
       => ? [I7: multiset(A),J6: multiset(A)] :
            ( ( N7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),J6) )
            & ? [K8: multiset(A)] :
                ( ( M6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),K8) )
                & ( J6 != zero_zero(multiset(A)) )
                & ! [X5: A] :
                    ( pp(aa(set(A),bool,member(A,X5),aa(multiset(A),set(A),set_mset(A),K8)))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),aa(multiset(A),set(A),set_mset(A),J6)))
                        & pp(aa(A,bool,aa(A,fun(A,bool),R3,X5),Xa4)) ) ) ) ) ) ) ).

% multp_implies_one_step
tff(fact_8112_Quotient__multiset,axiom,
    ! [A: $tType] : quotient(fun(A,nat),multiset(A),bNF_eq_onp(fun(A,nat),aTP_Lamp_akl(fun(A,nat),bool)),abs_multiset(A),count(A),cr_multiset(A)) ).

% Quotient_multiset
tff(fact_8113_Quotient__eq__onp__typedef,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),Abs: fun(A,B),Rep: fun(B,A),Cr: fun(A,fun(B,bool))] :
      ( quotient(A,B,bNF_eq_onp(A,P2),Abs,Rep,Cr)
     => type_definition(B,A,Rep,Abs,aa(fun(A,bool),set(A),collect(A),P2)) ) ).

% Quotient_eq_onp_typedef
tff(fact_8114_cr__multiset__def,axiom,
    ! [A: $tType,X5: fun(A,nat),Xa: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(fun(A,nat),fun(multiset(A),bool),cr_multiset(A),X5),Xa))
    <=> ( X5 = aa(multiset(A),fun(A,nat),count(A),Xa) ) ) ).

% cr_multiset_def
tff(fact_8115_Quotient__crel__typecopy,axiom,
    ! [A: $tType,B: $tType,Abs: fun(A,B),Rep: fun(B,A),T8: fun(A,fun(B,bool))] :
      ( quotient(A,B,fequal(A),Abs,Rep,T8)
     => ! [X5: A,Xa: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X5),Xa))
        <=> ( X5 = aa(B,A,Rep,Xa) ) ) ) ).

% Quotient_crel_typecopy
tff(fact_8116_Quotient__crel__typedef,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),Abs: fun(A,B),Rep: fun(B,A),T8: fun(A,fun(B,bool))] :
      ( quotient(A,B,bNF_eq_onp(A,P2),Abs,Rep,T8)
     => ! [X5: A,Xa: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),T8,X5),Xa))
        <=> ( X5 = aa(B,A,Rep,Xa) ) ) ) ).

% Quotient_crel_typedef
tff(fact_8117_multeqp__code__eq__reflclp__multp,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] :
      ( irreflp(A,R3)
     => ( transp(A,R3)
       => ( multeqp_code(A,R3) = aa(fun(multiset(A),fun(multiset(A),bool)),fun(multiset(A),fun(multiset(A),bool)),aa(fun(multiset(A),fun(multiset(A),bool)),fun(fun(multiset(A),fun(multiset(A),bool)),fun(multiset(A),fun(multiset(A),bool))),sup_sup(fun(multiset(A),fun(multiset(A),bool))),multp(A,R3)),fequal(multiset(A))) ) ) ) ).

% multeqp_code_eq_reflclp_multp
tff(fact_8118_multp__cancel__max,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),X6: multiset(A),Y6: multiset(A)] :
      ( transp(A,R3)
     => ( irreflp(A,R3)
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),X6),Y6))
        <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X6),Y6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y6),X6))) ) ) ) ).

% multp_cancel_max
tff(fact_8119_irreflp__multp,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] :
      ( transp(A,R3)
     => ( irreflp(A,R3)
       => irreflp(multiset(A),multp(A,R3)) ) ) ).

% irreflp_multp
tff(fact_8120_irreflp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,ord_less(A)) ) ).

% irreflp_less
tff(fact_8121_irreflp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,aTP_Lamp_ax(A,fun(A,bool))) ) ).

% irreflp_greater
tff(fact_8122_subset__mset_Oirreflp__greater,axiom,
    ! [A: $tType] : irreflp(multiset(A),aTP_Lamp_atw(multiset(A),fun(multiset(A),bool))) ).

% subset_mset.irreflp_greater
tff(fact_8123_irreflp__irrefl__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irreflp(A,aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),R2))
    <=> irrefl(A,R2) ) ).

% irreflp_irrefl_eq
tff(fact_8124_multp__cancel__add__mset,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),Uu: A,X6: multiset(A),Y6: multiset(A)] :
      ( transp(A,R3)
     => ( irreflp(A,R3)
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),X6)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),Y6)))
        <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),X6),Y6)) ) ) ) ).

% multp_cancel_add_mset
tff(fact_8125_multp__cancel,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),X6: multiset(A),Z5: multiset(A),Y6: multiset(A)] :
      ( transp(A,R3)
     => ( irreflp(A,R3)
       => ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X6),Z5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y6),Z5)))
        <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),multp(A,R3),X6),Y6)) ) ) ) ).

% multp_cancel
tff(fact_8126_prod__set__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : basic_snds(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Y),bot_bot(set(B))) ).

% prod_set_simps(2)
tff(fact_8127_prod__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B] : basic_fsts(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ).

% prod_set_simps(1)
tff(fact_8128_prod_Orel__compp__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool))] : basic_rel_prod(A,C,B,D,R1,R22) = aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),bool)),fun(product_prod(A,B),fun(product_prod(C,D),bool)),aa(fun(product_prod(A,B),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),fun(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),bool)),fun(product_prod(A,B),fun(product_prod(C,D),bool))),relcompp(product_prod(A,B),product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D)),conversep(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_aus(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R1),R22)),product_map_prod(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D))))),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_aus(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R1),R22)),product_map_prod(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)))) ).

% prod.rel_compp_Grp
tff(fact_8129_prod_Oin__rel,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool)),A3: product_prod(A,B),B2: product_prod(C,D)] :
      ( pp(aa(product_prod(C,D),bool,aa(product_prod(A,B),fun(product_prod(C,D),bool),basic_rel_prod(A,C,B,D,R1,R22),A3),B2))
    <=> ? [Z2: product_prod(product_prod(A,C),product_prod(B,D))] :
          ( pp(aa(set(product_prod(product_prod(A,C),product_prod(B,D))),bool,member(product_prod(product_prod(A,C),product_prod(B,D)),Z2),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_aus(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R1),R22))))
          & ( aa(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),product_map_prod(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D)),Z2) = A3 )
          & ( aa(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),product_map_prod(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)),Z2) = B2 ) ) ) ).

% prod.in_rel
tff(fact_8130_rel__prod__inject,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),A3: A,B2: C,C3: B,D3: D] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C3),D3)))
    <=> ( pp(aa(B,bool,aa(A,fun(B,bool),R1,A3),C3))
        & pp(aa(D,bool,aa(C,fun(D,bool),R22,B2),D3)) ) ) ).

% rel_prod_inject
tff(fact_8131_Pair__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] : pp(aa(fun(B,fun(D,product_prod(B,D))),bool,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),bool),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),A6,bNF_rel_fun(C,D,product_prod(A,C),product_prod(B,D),B5,basic_rel_prod(A,B,C,D,A6,B5))),product_Pair(A,C)),product_Pair(B,D))) ).

% Pair_transfer
tff(fact_8132_rat_Odomain,axiom,
    ! [X5: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(fun(product_prod(int,int),fun(rat,bool)),fun(product_prod(int,int),bool),domainp(product_prod(int,int),rat),pcr_rat),X5))
    <=> ? [Y4: product_prod(int,int)] :
          ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),X5),Y4))
          & pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Y4),Y4)) ) ) ).

% rat.domain
tff(fact_8133_prod_Orel__mono,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,bool)),R1a: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool)),R2a: fun(B,fun(D,bool))] :
      ( pp(aa(fun(A,fun(C,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(A,fun(C,bool)),bool),ord_less_eq(fun(A,fun(C,bool))),R1),R1a))
     => ( pp(aa(fun(B,fun(D,bool)),bool,aa(fun(B,fun(D,bool)),fun(fun(B,fun(D,bool)),bool),ord_less_eq(fun(B,fun(D,bool))),R22),R2a))
       => pp(aa(fun(product_prod(A,B),fun(product_prod(C,D),bool)),bool,aa(fun(product_prod(A,B),fun(product_prod(C,D),bool)),fun(fun(product_prod(A,B),fun(product_prod(C,D),bool)),bool),ord_less_eq(fun(product_prod(A,B),fun(product_prod(C,D),bool))),basic_rel_prod(A,C,B,D,R1,R22)),basic_rel_prod(A,C,B,D,R1a,R2a))) ) ) ).

% prod.rel_mono
tff(fact_8134_int_Oid__abs__transfer,axiom,
    pp(aa(fun(product_prod(nat,nat),int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),int),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),int,basic_rel_prod(nat,nat,nat,nat,fequal(nat),fequal(nat)),pcr_int),aTP_Lamp_aut(product_prod(nat,nat),product_prod(nat,nat))),abs_Integ)) ).

% int.id_abs_transfer
tff(fact_8135_rel__prod_Ocases,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),A1),A22))
     => ~ ! [A5: A,B4: B,C2: C] :
            ( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),C2) )
           => ! [D2: D] :
                ( ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B4),D2) )
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R1,A5),B4))
                 => ~ pp(aa(D,bool,aa(C,fun(D,bool),R22,C2),D2)) ) ) ) ) ).

% rel_prod.cases
tff(fact_8136_rel__prod_Osimps,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),A1),A22))
    <=> ? [A9: A,B7: B,C5: C,D6: D] :
          ( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A9),C5) )
          & ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B7),D6) )
          & pp(aa(B,bool,aa(A,fun(B,bool),R1,A9),B7))
          & pp(aa(D,bool,aa(C,fun(D,bool),R22,C5),D6)) ) ) ).

% rel_prod.simps
tff(fact_8137_rel__prod_Ointros,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,R1: fun(A,fun(B,bool)),A3: A,B2: B,R22: fun(C,fun(D,bool)),C3: C,D3: D] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R1,A3),B2))
     => ( pp(aa(D,bool,aa(C,fun(D,bool),R22,C3),D3))
       => pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C3)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),D3))) ) ) ).

% rel_prod.intros
tff(fact_8138_prod_Orel__map_I1_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,S1b: fun(E,fun(C,bool)),S2b: fun(F2,fun(D,bool)),I1: fun(A,E),I22: fun(B,F2),X: product_prod(A,B),Y: product_prod(C,D)] :
      ( pp(aa(product_prod(C,D),bool,aa(product_prod(E,F2),fun(product_prod(C,D),bool),basic_rel_prod(E,C,F2,D,S1b,S2b),aa(product_prod(A,B),product_prod(E,F2),product_map_prod(A,E,B,F2,I1,I22),X)),Y))
    <=> pp(aa(product_prod(C,D),bool,aa(product_prod(A,B),fun(product_prod(C,D),bool),basic_rel_prod(A,C,B,D,aa(fun(A,E),fun(A,fun(C,bool)),aTP_Lamp_auu(fun(E,fun(C,bool)),fun(fun(A,E),fun(A,fun(C,bool))),S1b),I1),aa(fun(B,F2),fun(B,fun(D,bool)),aTP_Lamp_auv(fun(F2,fun(D,bool)),fun(fun(B,F2),fun(B,fun(D,bool))),S2b),I22)),X),Y)) ) ).

% prod.rel_map(1)
tff(fact_8139_prod_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,S1a: fun(A,fun(E,bool)),S2a: fun(B,fun(F2,bool)),X: product_prod(A,B),G1: fun(C,E),G22: fun(D,F2),Y: product_prod(C,D)] :
      ( pp(aa(product_prod(E,F2),bool,aa(product_prod(A,B),fun(product_prod(E,F2),bool),basic_rel_prod(A,E,B,F2,S1a,S2a),X),aa(product_prod(C,D),product_prod(E,F2),product_map_prod(C,E,D,F2,G1,G22),Y)))
    <=> pp(aa(product_prod(C,D),bool,aa(product_prod(A,B),fun(product_prod(C,D),bool),basic_rel_prod(A,C,B,D,aa(fun(C,E),fun(A,fun(C,bool)),aTP_Lamp_auw(fun(A,fun(E,bool)),fun(fun(C,E),fun(A,fun(C,bool))),S1a),G1),aa(fun(D,F2),fun(B,fun(D,bool)),aTP_Lamp_aux(fun(B,fun(F2,bool)),fun(fun(D,F2),fun(B,fun(D,bool))),S2a),G22)),X),Y)) ) ).

% prod.rel_map(2)
tff(fact_8140_rel__prod__conv,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool))] : basic_rel_prod(A,C,B,D,R1,R22) = aa(fun(A,fun(B,fun(product_prod(C,D),bool))),fun(product_prod(A,B),fun(product_prod(C,D),bool)),product_case_prod(A,B,fun(product_prod(C,D),bool)),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool))),aTP_Lamp_auz(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool)))),R1),R22)) ).

% rel_prod_conv
tff(fact_8141_curry__transfer,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: $tType,E: $tType,D: $tType,A6: fun(A,fun(D,bool)),B5: fun(B,fun(E,bool)),C4: fun(C,fun(F2,bool))] : pp(aa(fun(fun(product_prod(D,E),F2),fun(D,fun(E,F2))),bool,aa(fun(fun(product_prod(A,B),C),fun(A,fun(B,C))),fun(fun(fun(product_prod(D,E),F2),fun(D,fun(E,F2))),bool),bNF_rel_fun(fun(product_prod(A,B),C),fun(product_prod(D,E),F2),fun(A,fun(B,C)),fun(D,fun(E,F2)),bNF_rel_fun(product_prod(A,B),product_prod(D,E),C,F2,basic_rel_prod(A,D,B,E,A6,B5),C4),bNF_rel_fun(A,D,fun(B,C),fun(E,F2),A6,bNF_rel_fun(B,E,C,F2,B5,C4))),product_curry(A,B,C)),product_curry(D,E,F2))) ).

% curry_transfer
tff(fact_8142_rat_Odomain__par__left__total,axiom,
    ! [P4: fun(product_prod(int,int),bool)] :
      ( left_total(product_prod(int,int),product_prod(int,int),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)))
     => ( pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal(bool)),P4),aTP_Lamp_ava(product_prod(int,int),bool)))
       => ( aa(fun(product_prod(int,int),fun(rat,bool)),fun(product_prod(int,int),bool),domainp(product_prod(int,int),rat),pcr_rat) = P4 ) ) ) ).

% rat.domain_par_left_total
tff(fact_8143_prod_Orel__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A15: set(A),F1: fun(A,C),A24: set(B),F22: fun(B,D)] : basic_rel_prod(A,C,B,D,bNF_Grp(A,C,A15,F1),bNF_Grp(B,D,A24,F22)) = bNF_Grp(product_prod(A,B),product_prod(C,D),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(B),fun(product_prod(A,B),bool),aTP_Lamp_avb(set(A),fun(set(B),fun(product_prod(A,B),bool)),A15),A24)),product_map_prod(A,C,B,D,F1,F22)) ).

% prod.rel_Grp
tff(fact_8144_rat_Odomain__par,axiom,
    ! [DR1: fun(int,bool),DR2: fun(int,bool),P23: fun(product_prod(int,int),bool)] :
      ( ( aa(fun(int,fun(int,bool)),fun(int,bool),domainp(int,int),fequal(int)) = DR1 )
     => ( ( aa(fun(int,fun(int,bool)),fun(int,bool),domainp(int,int),fequal(int)) = DR2 )
       => ( pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal(bool)),P23),aTP_Lamp_ava(product_prod(int,int),bool)))
         => ( aa(fun(product_prod(int,int),fun(rat,bool)),fun(product_prod(int,int),bool),domainp(product_prod(int,int),rat),pcr_rat) = aa(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),inf_inf(fun(product_prod(int,int),bool)),basic_pred_prod(int,int,DR1,DR2)),P23) ) ) ) ) ).

% rat.domain_par
tff(fact_8145_single__valuedp__single__valued__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] :
      ( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),R3))
    <=> single_valued(A,B,R3) ) ).

% single_valuedp_single_valued_eq
tff(fact_8146_pred__prod__inject,axiom,
    ! [A: $tType,B: $tType,P1: fun(A,bool),P24: fun(B,bool),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P1,P24),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
    <=> ( pp(aa(A,bool,P1,A3))
        & pp(aa(B,bool,P24,B2)) ) ) ).

% pred_prod_inject
tff(fact_8147_prod_Omap__cong__pred,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod(A,B),Ya: product_prod(A,B),F1: fun(A,C),G1: fun(A,C),F22: fun(B,D),G22: fun(B,D)] :
      ( ( X = Ya )
     => ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,aa(fun(A,C),fun(A,bool),aTP_Lamp_avc(fun(A,C),fun(fun(A,C),fun(A,bool)),F1),G1),aa(fun(B,D),fun(B,bool),aTP_Lamp_avd(fun(B,D),fun(fun(B,D),fun(B,bool)),F22),G22)),Ya))
       => ( aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),X) = aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,G1,G22),Ya) ) ) ) ).

% prod.map_cong_pred
tff(fact_8148_prod_Opred__True,axiom,
    ! [B: $tType,A: $tType,X5: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,aTP_Lamp_av(A,bool),aTP_Lamp_ave(B,bool)),X5)) ).

% prod.pred_True
tff(fact_8149_pred__prod_Ocases,axiom,
    ! [A: $tType,B: $tType,P1: fun(A,bool),P24: fun(B,bool),A3: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P1,P24),A3))
     => ~ ! [A5: A,B4: B] :
            ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
           => ( pp(aa(A,bool,P1,A5))
             => ~ pp(aa(B,bool,P24,B4)) ) ) ) ).

% pred_prod.cases
tff(fact_8150_pred__prod_Osimps,axiom,
    ! [A: $tType,B: $tType,P1: fun(A,bool),P24: fun(B,bool),A3: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P1,P24),A3))
    <=> ? [A9: A,B7: B] :
          ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B7) )
          & pp(aa(A,bool,P1,A9))
          & pp(aa(B,bool,P24,B7)) ) ) ).

% pred_prod.simps
tff(fact_8151_pred__prod_Ointros,axiom,
    ! [A: $tType,B: $tType,P1: fun(A,bool),A3: A,P24: fun(B,bool),B2: B] :
      ( pp(aa(A,bool,P1,A3))
     => ( pp(aa(B,bool,P24,B2))
       => pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P1,P24),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ) ).

% pred_prod.intros
tff(fact_8152_pred__prod__split,axiom,
    ! [A: $tType,B: $tType,P2: fun(bool,bool),Q2: fun(A,bool),R2: fun(B,bool),Xy: product_prod(A,B)] :
      ( pp(aa(bool,bool,P2,aa(product_prod(A,B),bool,basic_pred_prod(A,B,Q2,R2),Xy)))
    <=> ! [X4: A,Y4: B] :
          ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) )
         => pp(aa(bool,bool,P2,fconj(aa(A,bool,Q2,X4),aa(B,bool,R2,Y4)))) ) ) ).

% pred_prod_split
tff(fact_8153_prod_Opred__mono,axiom,
    ! [A: $tType,B: $tType,P1: fun(A,bool),P1a: fun(A,bool),P24: fun(B,bool),P2a: fun(B,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P1),P1a))
     => ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),P24),P2a))
       => pp(aa(fun(product_prod(A,B),bool),bool,aa(fun(product_prod(A,B),bool),fun(fun(product_prod(A,B),bool),bool),ord_less_eq(fun(product_prod(A,B),bool)),basic_pred_prod(A,B,P1,P24)),basic_pred_prod(A,B,P1a,P2a))) ) ) ).

% prod.pred_mono
tff(fact_8154_single__valuedp__less__eq,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),S2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R3),S2))
     => ( single_valuedp(A,B,S2)
       => single_valuedp(A,B,R3) ) ) ).

% single_valuedp_less_eq
tff(fact_8155_single__valuedp__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
      ( single_valuedp(A,B,R3)
    <=> ! [X4: A] : uniq(B,aa(A,fun(B,bool),R3,X4)) ) ).

% single_valuedp_iff_Uniq
tff(fact_8156_Abs__rat__cases,axiom,
    ! [X: rat] :
      ~ ! [Y3: set(product_prod(int,int))] :
          ( ( X = aa(set(product_prod(int,int)),rat,abs_rat,Y3) )
         => ~ pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y3),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool)))) ) ).

% Abs_rat_cases
tff(fact_8157_Abs__rat__induct,axiom,
    ! [P2: fun(rat,bool),X: rat] :
      ( ! [Y3: set(product_prod(int,int))] :
          ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y3),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool))))
         => pp(aa(rat,bool,P2,aa(set(product_prod(int,int)),rat,abs_rat,Y3))) )
     => pp(aa(rat,bool,P2,X)) ) ).

% Abs_rat_induct
tff(fact_8158_Abs__rat__inject,axiom,
    ! [X: set(product_prod(int,int)),Y: set(product_prod(int,int))] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),X),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool))))
     => ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool))))
       => ( ( aa(set(product_prod(int,int)),rat,abs_rat,X) = aa(set(product_prod(int,int)),rat,abs_rat,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_rat_inject
tff(fact_8159_Abs__rat__inverse,axiom,
    ! [Y: set(product_prod(int,int))] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool))))
     => ( aa(rat,set(product_prod(int,int)),rep_rat,aa(set(product_prod(int,int)),rat,abs_rat,Y)) = Y ) ) ).

% Abs_rat_inverse
tff(fact_8160_type__definition__rat,axiom,
    type_definition(rat,set(product_prod(int,int)),rep_rat,abs_rat,aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool))) ).

% type_definition_rat
tff(fact_8161_Rep__rat,axiom,
    ! [X: rat] : pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),aa(rat,set(product_prod(int,int)),rep_rat,X)),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool)))) ).

% Rep_rat
tff(fact_8162_Rep__rat__cases,axiom,
    ! [Y: set(product_prod(int,int))] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool))))
     => ~ ! [X3: rat] : Y != aa(rat,set(product_prod(int,int)),rep_rat,X3) ) ).

% Rep_rat_cases
tff(fact_8163_Rep__rat__induct,axiom,
    ! [Y: set(product_prod(int,int)),P2: fun(set(product_prod(int,int)),bool)] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_avf(set(product_prod(int,int)),bool))))
     => ( ! [X3: rat] : pp(aa(set(product_prod(int,int)),bool,P2,aa(rat,set(product_prod(int,int)),rep_rat,X3)))
       => pp(aa(set(product_prod(int,int)),bool,P2,Y)) ) ) ).

% Rep_rat_induct
tff(fact_8164_Partial__order__Restr,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A6: set(A)] :
      ( order_7125193373082350890der_on(A,field2(A,R3),R3)
     => order_7125193373082350890der_on(A,field2(A,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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))),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))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_abo(set(A),fun(A,set(A)),A6)))) ) ).

% Partial_order_Restr
tff(fact_8165_one__assn__def,axiom,
    one_one(assn) = abs_assn(one_assn_raw) ).

% one_assn_def
tff(fact_8166_natLeq__Partial__order,axiom,
    order_7125193373082350890der_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Partial_order
tff(fact_8167_one__assn__raw_Osimps,axiom,
    ! [H: heap_ext(product_unit),As: set(nat)] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))
    <=> ( As = bot_bot(set(nat)) ) ) ).

% one_assn_raw.simps
tff(fact_8168_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
      <=> pp(Y) )
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ( pp(Y)
            <=> ( As2 != bot_bot(set(nat)) ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_8169_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ( As2 != bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_8170_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
            ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
           => ( As2 = bot_bot(set(nat)) ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_8171_Zorns__po__lemma,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,field2(A,R3),R3)
     => ( ! [C8: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),C8),chains(A,R3)))
           => ? [X5: A] :
                ( pp(aa(set(A),bool,member(A,X5),field2(A,R3)))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,member(A,Xa4),C8))
                   => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X5)),R3)) ) ) )
       => ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),field2(A,R3)))
            & ! [Xa: A] :
                ( pp(aa(set(A),bool,member(A,Xa),field2(A,R3)))
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa)),R3))
                 => ( Xa = X3 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_8172_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
               => ( As2 = bot_bot(set(nat)) ) ) ) ) ) ).

% one_assn_raw.pelims(3)
tff(fact_8173_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2)))
               => ( As2 != bot_bot(set(nat)) ) ) ) ) ) ).

% one_assn_raw.pelims(2)
tff(fact_8174_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
      ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,one_assn_raw,X))
      <=> pp(Y) )
     => ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X))
       => ~ ! [H2: heap_ext(product_unit),As2: set(nat)] :
              ( ( X = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2) )
             => ( ( pp(Y)
                <=> ( As2 = bot_bot(set(nat)) ) )
               => ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As2))) ) ) ) ) ).

% one_assn_raw.pelims(1)
tff(fact_8175_acyclic__insert,axiom,
    ! [A: $tType,Y: A,X: A,R3: set(product_prod(A,A))] :
      ( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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),Y),X)),R3))
    <=> ( transitive_acyclic(A,R3)
        & ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R3))) ) ) ).

% acyclic_insert
tff(fact_8176_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B72)),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B62))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B52))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B42))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B32))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B22))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B1))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B0)) ).

% integer_of_char_code
tff(fact_8177_acyclic__empty,axiom,
    ! [A: $tType] : transitive_acyclic(A,bot_bot(set(product_prod(A,A)))) ).

% acyclic_empty
tff(fact_8178_acyclicP__converse,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] :
      ( transitive_acyclic(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),conversep(A,A,R3))))
    <=> transitive_acyclic(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R3))) ) ).

% acyclicP_converse
tff(fact_8179_acyclicI__order,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [R3: set(product_prod(B,B)),F3: fun(B,A)] :
          ( ! [A5: B,B4: B] :
              ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B4)),R3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,B4)),aa(B,A,F3,A5))) )
         => transitive_acyclic(B,R3) ) ) ).

% acyclicI_order
tff(fact_8180_cyclicE,axiom,
    ! [A: $tType,G3: set(product_prod(A,A))] :
      ( ~ transitive_acyclic(A,G3)
     => ~ ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),transitive_trancl(A,G3))) ) ).

% cyclicE
tff(fact_8181_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_acp(product_prod(int,int),product_prod(int,int)),Uu) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),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(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ).

% ATP.lambda_1
tff(fact_8182_ATP_Olambda__2,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_yc(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,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),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_2
tff(fact_8183_ATP_Olambda__3,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aa(nat,bool),Uu))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),lim(product_unit,h2)))
        & ~ pp(aa(set(nat),bool,member(nat,Uu),hoare_new_addrs(h3,as,h2))) ) ) ).

% ATP.lambda_3
tff(fact_8184_ATP_Olambda__4,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ab(nat,bool),Uu))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),lim(product_unit,h3)))
        & ~ pp(aa(set(nat),bool,member(nat,Uu),as)) ) ) ).

% ATP.lambda_4
tff(fact_8185_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_xy(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(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_5
tff(fact_8186_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_aua(A,bool),Uu))
        <=> ( pp(aa(set(A),bool,member(A,Uu),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Uu)) ) ) ) ).

% ATP.lambda_6
tff(fact_8187_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_aeq(product_prod(int,int),A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_7
tff(fact_8188_ATP_Olambda__8,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_acn(product_prod(int,int),product_prod(int,int)),Uu) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(product_prod(int,int),int,product_snd(int,int),Uu)) ).

% ATP.lambda_8
tff(fact_8189_ATP_Olambda__9,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_ow(product_prod(A,A),bool),Uu))
    <=> ( aa(product_prod(A,A),A,product_fst(A,A),Uu) = aa(product_prod(A,A),A,product_snd(A,A),Uu) ) ) ).

% ATP.lambda_9
tff(fact_8190_ATP_Olambda__10,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_ajj(list(A),product_prod(nat,list(A))),Uu) = aa(list(A),product_prod(nat,list(A)),aa(nat,fun(list(A),product_prod(nat,list(A))),product_Pair(nat,list(A)),aa(list(A),nat,size_size(list(A)),Uu)),Uu) ).

% ATP.lambda_10
tff(fact_8191_ATP_Olambda__11,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( pp(aa(nat,bool,aTP_Lamp_ahd(nat,bool),Uu))
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_11
tff(fact_8192_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_gt(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_12
tff(fact_8193_ATP_Olambda__13,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_abl(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).

% ATP.lambda_13
tff(fact_8194_ATP_Olambda__14,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_ng(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).

% ATP.lambda_14
tff(fact_8195_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_ox(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_15
tff(fact_8196_ATP_Olambda__16,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_ava(product_prod(int,int),bool),Uu))
    <=> pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Uu),Uu)) ) ).

% ATP.lambda_16
tff(fact_8197_ATP_Olambda__17,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_da(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_17
tff(fact_8198_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_acr(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uu),zero_zero(nat)) ).

% ATP.lambda_18
tff(fact_8199_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu: A] : aa(A,multiset(A),aTP_Lamp_ait(A,multiset(A)),Uu) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),zero_zero(multiset(A))) ).

% ATP.lambda_19
tff(fact_8200_ATP_Olambda__20,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,list(A),aTP_Lamp_sq(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).

% ATP.lambda_20
tff(fact_8201_ATP_Olambda__21,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_wk(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_21
tff(fact_8202_ATP_Olambda__22,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_wt(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_22
tff(fact_8203_ATP_Olambda__23,axiom,
    ! [B: $tType,D: $tType,Uu: fun(D,B)] : aa(fun(D,B),set(B),aTP_Lamp_ann(fun(D,B),set(B)),Uu) = aa(set(D),set(B),image2(D,B,Uu),top_top(set(D))) ).

% ATP.lambda_23
tff(fact_8204_ATP_Olambda__24,axiom,
    ! [A: $tType,D: $tType,Uu: fun(D,A)] : aa(fun(D,A),set(A),aTP_Lamp_anm(fun(D,A),set(A)),Uu) = aa(set(D),set(A),image2(D,A,Uu),top_top(set(D))) ).

% ATP.lambda_24
tff(fact_8205_ATP_Olambda__25,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_ajn(list(B),bool),Uu))
    <=> ( Uu = nil(B) ) ) ).

% ATP.lambda_25
tff(fact_8206_ATP_Olambda__26,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_ajm(list(A),bool),Uu))
    <=> ( Uu = nil(A) ) ) ).

% ATP.lambda_26
tff(fact_8207_ATP_Olambda__27,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_acq(product_prod(int,int),bool),Uu))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))) ) ).

% ATP.lambda_27
tff(fact_8208_ATP_Olambda__28,axiom,
    ! [Uu: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_ez(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aTP_Lamp_ey(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu)) ).

% ATP.lambda_28
tff(fact_8209_ATP_Olambda__29,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_qz(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_29
tff(fact_8210_ATP_Olambda__30,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_uu(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = aa(product_prod(A,A),fun(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)) ).

% ATP.lambda_30
tff(fact_8211_ATP_Olambda__31,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_ra(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_31
tff(fact_8212_ATP_Olambda__32,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_rf(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_32
tff(fact_8213_ATP_Olambda__33,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_atd(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_atc(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_33
tff(fact_8214_ATP_Olambda__34,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_uk(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_uj(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_34
tff(fact_8215_ATP_Olambda__35,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_ss(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_sr(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_35
tff(fact_8216_ATP_Olambda__36,axiom,
    ! [A: $tType,Uu: multiset(A)] : aa(multiset(A),set(A),aTP_Lamp_akr(multiset(A),set(A)),Uu) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aki(multiset(A),fun(A,bool),Uu)) ).

% ATP.lambda_36
tff(fact_8217_ATP_Olambda__37,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_ahc(nat,set(nat)),Uu) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_lh(nat,fun(nat,bool),Uu)) ).

% ATP.lambda_37
tff(fact_8218_ATP_Olambda__38,axiom,
    ! [B: $tType,Uu: fun(B,nat)] :
      ( pp(aa(fun(B,nat),bool,aTP_Lamp_apc(fun(B,nat),bool),Uu))
    <=> pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_aoy(fun(B,nat),fun(B,bool),Uu)))) ) ).

% ATP.lambda_38
tff(fact_8219_ATP_Olambda__39,axiom,
    ! [A: $tType,Uu: fun(A,nat)] :
      ( pp(aa(fun(A,nat),bool,aTP_Lamp_akl(fun(A,nat),bool),Uu))
    <=> pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_kx(fun(A,nat),fun(A,bool),Uu)))) ) ).

% ATP.lambda_39
tff(fact_8220_ATP_Olambda__40,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_aho(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ahn(fun(A,B),fun(A,fun(B,bool)),Uu))) ).

% ATP.lambda_40
tff(fact_8221_ATP_Olambda__41,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_qy(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_41
tff(fact_8222_ATP_Olambda__42,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_re(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_42
tff(fact_8223_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] : aa(nat,heap_Time_Heap(A),aTP_Lamp_df(nat,heap_Time_Heap(A)),Uu) = aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_43
tff(fact_8224_ATP_Olambda__44,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_es(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_44
tff(fact_8225_ATP_Olambda__45,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_dz(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_45
tff(fact_8226_ATP_Olambda__46,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_oc(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_46
tff(fact_8227_ATP_Olambda__47,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_od(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_47
tff(fact_8228_ATP_Olambda__48,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_dy(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_48
tff(fact_8229_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ahy(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_49
tff(fact_8230_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ahx(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_50
tff(fact_8231_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_agk(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_51
tff(fact_8232_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_agj(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_52
tff(fact_8233_ATP_Olambda__53,axiom,
    ! [Uu: int] : aa(int,nat,aTP_Lamp_ahe(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_53
tff(fact_8234_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_atz(A,bool),Uu))
        <=> ? [N3: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N3) )
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N3)) ) ) ) ).

% ATP.lambda_54
tff(fact_8235_ATP_Olambda__55,axiom,
    ! [Uu: set(product_prod(int,int))] :
      ( pp(aa(set(product_prod(int,int)),bool,aTP_Lamp_avf(set(product_prod(int,int)),bool),Uu))
    <=> ? [X4: product_prod(int,int)] :
          ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X4),X4))
          & ( Uu = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X4)) ) ) ) ).

% ATP.lambda_55
tff(fact_8236_ATP_Olambda__56,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_adi(product_prod(A,A),bool),Uu))
    <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% ATP.lambda_56
tff(fact_8237_ATP_Olambda__57,axiom,
    ! [A: $tType,B: $tType,Uu: product_prod(product_prod(bool,A),product_prod(bool,B))] :
      ( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_aee(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu))
    <=> ? [X4: A,Y4: B] : Uu = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fTrue),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y4)) ) ).

% ATP.lambda_57
tff(fact_8238_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A)] : aa(array(A),heap_Time_Heap(product_unit),aTP_Lamp_nz(array(A),heap_Time_Heap(product_unit)),Uu) = aa(product_unit,heap_Time_Heap(product_unit),heap_Time_ureturn(product_unit),product_Unity) ) ).

% ATP.lambda_58
tff(fact_8239_ATP_Olambda__59,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ff(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one),aTP_Lamp_fd(nat,fun(num,option(num)),Uua),aTP_Lamp_fe(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_59
tff(fact_8240_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gk(nat,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_60
tff(fact_8241_ATP_Olambda__61,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] : aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_tz(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(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),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_61
tff(fact_8242_ATP_Olambda__62,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,option(A),aTP_Lamp_nb(list(A),fun(nat,option(A)),Uu),Uua) = if(option(A),aa(nat,bool,aa(nat,fun(nat,bool),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_62
tff(fact_8243_ATP_Olambda__63,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_fk(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(code_integer,int,code_int_of_integer,Uu)),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),aa(num,num,bit0,one))),aa(code_integer,int,code_int_of_integer,Uu))),one_one(int))) ).

% ATP.lambda_63
tff(fact_8244_ATP_Olambda__64,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_fj(code_integer,fun(code_integer,num)),Uu),Uua) = if(num,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu))),one)) ).

% ATP.lambda_64
tff(fact_8245_ATP_Olambda__65,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_gi(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one(nat))) ).

% ATP.lambda_65
tff(fact_8246_ATP_Olambda__66,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_oo(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_66
tff(fact_8247_ATP_Olambda__67,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_acm(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),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),Uu),Uua)) ).

% ATP.lambda_67
tff(fact_8248_ATP_Olambda__68,axiom,
    ! [A: $tType,Uu: set(fun(A,nat)),Uua: A] : aa(A,nat,aa(set(fun(A,nat)),fun(A,nat),aTP_Lamp_aku(set(fun(A,nat)),fun(A,nat)),Uu),Uua) = if(nat,aa(set(fun(A,nat)),bool,aa(set(fun(A,nat)),fun(set(fun(A,nat)),bool),fequal(set(fun(A,nat))),Uu),bot_bot(set(fun(A,nat)))),zero_zero(nat),aa(set(nat),nat,complete_Inf_Inf(nat),aa(set(fun(A,nat)),set(nat),image2(fun(A,nat),nat,aTP_Lamp_akm(A,fun(fun(A,nat),nat),Uua)),Uu))) ).

% ATP.lambda_68
tff(fact_8249_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gj(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_69
tff(fact_8250_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_air(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ).

% ATP.lambda_70
tff(fact_8251_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_afm(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_71
tff(fact_8252_ATP_Olambda__72,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_ajr(list(list(A)),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(list(list(A)),bool,aa(list(A),fun(list(list(A)),bool),list_all2(A,list(A),aTP_Lamp_ajq(A,fun(list(A),bool))),Uua),Uu)) ) ).

% ATP.lambda_72
tff(fact_8253_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_cs(option(A),fun(A,option(A)),Uu),Uua) = case_option(option(A),A,none(A),aTP_Lamp_cr(A,fun(A,option(A)),Uua),Uu) ) ).

% ATP.lambda_73
tff(fact_8254_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ay(option(A),fun(A,bool),Uu),Uua))
        <=> pp(case_option(bool,A,fTrue,aa(A,fun(A,bool),aTP_Lamp_ax(A,fun(A,bool)),Uua),Uu)) ) ) ).

% ATP.lambda_74
tff(fact_8255_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_fg(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_ff(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_75
tff(fact_8256_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aw(option(A),fun(A,bool),Uu),Uua))
        <=> pp(case_option(bool,A,fFalse,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_76
tff(fact_8257_ATP_Olambda__77,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_fd(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_dz(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_77
tff(fact_8258_ATP_Olambda__78,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_pb(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua))) ) ).

% ATP.lambda_78
tff(fact_8259_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_agb(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_79
tff(fact_8260_ATP_Olambda__80,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_tg(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_tf(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_80
tff(fact_8261_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jq(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_81
tff(fact_8262_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ha(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gz(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_82
tff(fact_8263_ATP_Olambda__83,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_acl(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = 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(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_83
tff(fact_8264_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hr(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,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_84
tff(fact_8265_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hs(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,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),Uua)) ) ).

% ATP.lambda_85
tff(fact_8266_ATP_Olambda__86,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_aco(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = 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(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_86
tff(fact_8267_ATP_Olambda__87,axiom,
    ! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
      ( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_arv(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(A2,field2(A2,Uua),Uua)
        & ? [X_12: fun(A,A2)] : bNF_Wellorder_embedS(A,A2,Uu,Uua,X_12) ) ) ).

% ATP.lambda_87
tff(fact_8268_ATP_Olambda__88,axiom,
    ! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
      ( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_ary(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(A2,field2(A2,Uua),Uua)
        & ? [X_12: fun(A,A2)] : pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,Uu,Uua),X_12)) ) ) ).

% ATP.lambda_88
tff(fact_8269_ATP_Olambda__89,axiom,
    ! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
      ( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_arx(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(A2,field2(A2,Uua),Uua)
        & ? [X_12: fun(A,A2)] : bNF_Wellorder_iso(A,A2,Uu,Uua,X_12) ) ) ).

% ATP.lambda_89
tff(fact_8270_ATP_Olambda__90,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_oq(rat,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu))
        & pp(aa(rat,bool,aa(rat,fun(rat,bool),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_90
tff(fact_8271_ATP_Olambda__91,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_na(set(A),fun(list(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu))
        & distinct(A,Uua) ) ) ).

% ATP.lambda_91
tff(fact_8272_ATP_Olambda__92,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_mz(set(A),fun(list(A),bool),Uu),Uua))
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
        & distinct(A,Uua) ) ) ).

% ATP.lambda_92
tff(fact_8273_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hw(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_93
tff(fact_8274_ATP_Olambda__94,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_aik(rat,fun(product_prod(int,int),bool),Uu),Uua))
    <=> ( ( Uu = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Uua)))
        & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).

% ATP.lambda_94
tff(fact_8275_ATP_Olambda__95,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ain(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),aa(set(A),set(A),image(A,A,Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% ATP.lambda_95
tff(fact_8276_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gx(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_96
tff(fact_8277_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gl(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_97
tff(fact_8278_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hh(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,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_98
tff(fact_8279_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_aor(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,Uua,Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),array_length(A,Uua,Uu)))) ) ).

% ATP.lambda_99
tff(fact_8280_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_dt(nat,fun(nat,bool)),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_100
tff(fact_8281_ATP_Olambda__101,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_yd(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_101
tff(fact_8282_ATP_Olambda__102,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_aal(set(option(A)),fun(option(A),bool),Uu),Uua))
    <=> ( pp(aa(set(option(A)),bool,member(option(A),Uua),Uu))
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_102
tff(fact_8283_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hk(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,power_power(nat,aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))) ).

% ATP.lambda_103
tff(fact_8284_ATP_Olambda__104,axiom,
    ! [Uu: set(int),Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_amz(set(int),fun(int,bool),Uu),Uua))
    <=> ( pp(aa(set(int),bool,member(int,Uua),Uu))
        & ! [X4: int] :
            ( pp(aa(set(int),bool,member(int,X4),Uu))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X4),Uua)) ) ) ) ).

% ATP.lambda_104
tff(fact_8285_ATP_Olambda__105,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_amo(set(set(A)),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,member(set(A),Uua),Uu))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
           => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),X4)) ) ) ) ).

% ATP.lambda_105
tff(fact_8286_ATP_Olambda__106,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_aow(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),Uu),Uua))
        & ! [A9: A,B7: A,C5: A] :
            ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),B7)),Uua))
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C5)),Uu)) )
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A9),B7)),Uu)) ) ) ) ).

% ATP.lambda_106
tff(fact_8287_ATP_Olambda__107,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_aji(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & chain_subset(A,Uua) ) ) ).

% ATP.lambda_107
tff(fact_8288_ATP_Olambda__108,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aao(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu))
        & pp(aa(set(A),bool,finite_finite2(A),Uua)) ) ) ).

% ATP.lambda_108
tff(fact_8289_ATP_Olambda__109,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_lz(set(A),fun(set(A),bool)),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uu),Uua))
        & pp(aa(set(A),bool,finite_finite2(A),Uua)) ) ) ).

% ATP.lambda_109
tff(fact_8290_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,Uu),Uua)),Uua) ).

% ATP.lambda_110
tff(fact_8291_ATP_Olambda__111,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gv(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_111
tff(fact_8292_ATP_Olambda__112,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gu(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_112
tff(fact_8293_ATP_Olambda__113,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_wl(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(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_113
tff(fact_8294_ATP_Olambda__114,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_ace(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(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),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_114
tff(fact_8295_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_li(A,fun(A,bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,member(A,Uua),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).

% ATP.lambda_115
tff(fact_8296_ATP_Olambda__116,axiom,
    ! [D: $tType,B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(D,product_prod(A,B))] :
      ( pp(aa(fun(D,product_prod(A,B)),bool,aTP_Lamp_aaw(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(D),set(product_prod(A,B)),image2(D,product_prod(A,B),Uua),top_top(set(D)))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_116
tff(fact_8297_ATP_Olambda__117,axiom,
    ! [A: $tType,Uu: fun(set(A),bool),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_afu(fun(set(A),bool),fun(set(A),bool)),Uu),Uua))
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A16: set(A),A9: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A9),A16) )
            & pp(aa(set(A),bool,Uu,A16)) ) ) ) ).

% ATP.lambda_117
tff(fact_8298_ATP_Olambda__118,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aTP_Lamp_ase(set(B),fun(fun(A,B),bool),Uu),Uua))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,Uua),top_top(set(A)))),Uu)) ) ).

% ATP.lambda_118
tff(fact_8299_ATP_Olambda__119,axiom,
    ! [D: $tType,A: $tType,Uu: set(A),Uua: fun(D,A)] :
      ( pp(aa(fun(D,A),bool,aTP_Lamp_auh(set(A),fun(fun(D,A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,Uua),top_top(set(D)))),Uu)) ) ).

% ATP.lambda_119
tff(fact_8300_ATP_Olambda__120,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ail(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))),Uu) ).

% ATP.lambda_120
tff(fact_8301_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( pp(aa(list(A),bool,aTP_Lamp_atq(set(A),fun(list(A),bool),Uu),Uua))
        <=> ( sorted_wrt(A,ord_less(A),Uua)
            & ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).

% ATP.lambda_121
tff(fact_8302_ATP_Olambda__122,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aau(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).

% ATP.lambda_122
tff(fact_8303_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aaq(nat,fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).

% ATP.lambda_123
tff(fact_8304_ATP_Olambda__124,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_aav(fun(C,product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(C),set(B),image2(C,B,aa(fun(C,product_prod(A,B)),fun(C,B),comp(product_prod(A,B),B,C,product_snd(A,B)),Uu)),top_top(set(C))) ).

% ATP.lambda_124
tff(fact_8305_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_je(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),aa(num,num,bit0,one))),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),aa(num,num,bit0,one))),Uua)))) ) ).

% ATP.lambda_125
tff(fact_8306_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hg(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),aa(num,num,bit0,one))),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),aa(num,num,bit0,one))),Uua)))) ) ).

% ATP.lambda_126
tff(fact_8307_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_127
tff(fact_8308_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_io(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_128
tff(fact_8309_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_iz(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_129
tff(fact_8310_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_kh(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,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_130
tff(fact_8311_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_he(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_131
tff(fact_8312_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(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_132
tff(fact_8313_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_133
tff(fact_8314_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alg(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y4: A] :
                ( pp(aa(A,bool,Uu,Y4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y4)) ) ) ) ) ).

% ATP.lambda_134
tff(fact_8315_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alh(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y4: A] :
                ( pp(aa(A,bool,Uu,Y4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Uua)) ) ) ) ) ).

% ATP.lambda_135
tff(fact_8316_ATP_Olambda__136,axiom,
    ! [A: $tType,Uu: fun(multiset(A),bool),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_alj(fun(multiset(A),bool),fun(multiset(A),bool),Uu),Uua))
    <=> ( pp(aa(multiset(A),bool,Uu,Uua))
        & ! [Y4: multiset(A)] :
            ( pp(aa(multiset(A),bool,Uu,Y4))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Uua),Y4)) ) ) ) ).

% ATP.lambda_136
tff(fact_8317_ATP_Olambda__137,axiom,
    ! [A: $tType,Uu: fun(multiset(A),bool),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_aly(fun(multiset(A),bool),fun(multiset(A),bool),Uu),Uua))
    <=> ( pp(aa(multiset(A),bool,Uu,Uua))
        & ! [Y4: multiset(A)] :
            ( pp(aa(multiset(A),bool,Uu,Y4))
           => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Y4),Uua)) ) ) ) ).

% ATP.lambda_137
tff(fact_8318_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jv(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_138
tff(fact_8319_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ju(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_139
tff(fact_8320_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_agf(fun(A,A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua)) ) ) ).

% ATP.lambda_140
tff(fact_8321_ATP_Olambda__141,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_ajk(fun(A,nat),fun(A,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(A,nat,Uu,Uua)),Uua) ).

% ATP.lambda_141
tff(fact_8322_ATP_Olambda__142,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_ahm(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).

% ATP.lambda_142
tff(fact_8323_ATP_Olambda__143,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_rp(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_143
tff(fact_8324_ATP_Olambda__144,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_xx(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_144
tff(fact_8325_ATP_Olambda__145,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_ld(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_145
tff(fact_8326_ATP_Olambda__146,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_lg(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_146
tff(fact_8327_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_lr(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ka(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,Uu,Uua)) ) ).

% ATP.lambda_147
tff(fact_8328_ATP_Olambda__148,axiom,
    ! [A: $tType,Uu: list(fun(A,nat)),Uua: A] : aa(A,list(nat),aTP_Lamp_apo(list(fun(A,nat)),fun(A,list(nat)),Uu),Uua) = aa(list(fun(A,nat)),list(nat),map(fun(A,nat),nat,aTP_Lamp_akm(A,fun(fun(A,nat),nat),Uua)),Uu) ).

% ATP.lambda_148
tff(fact_8329_ATP_Olambda__149,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_sp(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_so(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_149
tff(fact_8330_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_aet(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_150
tff(fact_8331_ATP_Olambda__151,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_dx(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_151
tff(fact_8332_ATP_Olambda__152,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aoj(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).

% ATP.lambda_152
tff(fact_8333_ATP_Olambda__153,axiom,
    ! [Uu: assn,Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_de(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,in_range,Uua))
        & ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uu),Uua)) ) ) ).

% ATP.lambda_153
tff(fact_8334_ATP_Olambda__154,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: multiset(product_prod(A,B))] :
      ( pp(aa(multiset(product_prod(A,B)),bool,aTP_Lamp_akg(fun(A,fun(B,bool)),fun(multiset(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(multiset(product_prod(A,B)),set(product_prod(A,B)),set_mset(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_154
tff(fact_8335_ATP_Olambda__155,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: list(product_prod(A,B))] :
      ( pp(aa(list(product_prod(A,B)),bool,aTP_Lamp_ajl(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_155
tff(fact_8336_ATP_Olambda__156,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(set(A))] : aa(set(set(A)),set(A),aTP_Lamp_apg(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uua)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))) ).

% ATP.lambda_156
tff(fact_8337_ATP_Olambda__157,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_apk(fun(A,fun(B,bool)),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),Uua)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))) ).

% ATP.lambda_157
tff(fact_8338_ATP_Olambda__158,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
          ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_sf(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uu)),aa(product_prod(A,B),A,product_fst(A,B),Uua))) ) ) ).

% ATP.lambda_158
tff(fact_8339_ATP_Olambda__159,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
          ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_sg(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),A,product_fst(A,B),Uu))) ) ) ).

% ATP.lambda_159
tff(fact_8340_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_nq(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_160
tff(fact_8341_ATP_Olambda__161,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_abj(num,fun(num,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Uu)),aa(num,int,numeral_numeral(int),Uua)) ).

% ATP.lambda_161
tff(fact_8342_ATP_Olambda__162,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_st(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua)),product_lists(A,Uu)) ).

% ATP.lambda_162
tff(fact_8343_ATP_Olambda__163,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_amx(list(A),fun(list(A),bool)),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_163
tff(fact_8344_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_ry(list(A),fun(list(A),bool),Uu),Uua))
    <=> ( mset(A,Uua) = mset(A,Uu) ) ) ).

% ATP.lambda_164
tff(fact_8345_ATP_Olambda__165,axiom,
    ! [A: $tType,Uu: set(A),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_aui(set(A),fun(multiset(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(multiset(A),set(A),set_mset(A),Uua)),Uu)) ) ).

% ATP.lambda_165
tff(fact_8346_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aoz(set(A),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)) ) ).

% ATP.lambda_166
tff(fact_8347_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hp(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_167
tff(fact_8348_ATP_Olambda__168,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_sh(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).

% ATP.lambda_168
tff(fact_8349_ATP_Olambda__169,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_tc(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)),Uu)) ) ).

% ATP.lambda_169
tff(fact_8350_ATP_Olambda__170,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_tw(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,aa(nat,nat,suc,Uua)),Uu)) ) ).

% ATP.lambda_170
tff(fact_8351_ATP_Olambda__171,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_apa(nat,fun(list(A),bool),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).

% ATP.lambda_171
tff(fact_8352_ATP_Olambda__172,axiom,
    ! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aTP_Lamp_atr(heap_ext(product_unit),fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),lim(product_unit,Uu)),one_one(nat)) ).

% ATP.lambda_172
tff(fact_8353_ATP_Olambda__173,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(set(product_prod(A,A))),aTP_Lamp_asq(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),Uu),Uua) = aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A))),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),set(set(product_prod(A,A)))),insert(set(product_prod(A,A))),Uu),bot_bot(set(set(product_prod(A,A)))))) ).

% ATP.lambda_173
tff(fact_8354_ATP_Olambda__174,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_dn(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),aa(num,num,bit0,one))),Uua)),one_one(A))) ) ).

% ATP.lambda_174
tff(fact_8355_ATP_Olambda__175,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_do(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),aa(num,num,bit0,one))),Uua)) ) ).

% ATP.lambda_175
tff(fact_8356_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hy(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),aa(num,num,bit0,one)))) ) ).

% ATP.lambda_176
tff(fact_8357_ATP_Olambda__177,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_wg(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)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu)),Uua)) ).

% ATP.lambda_177
tff(fact_8358_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_qd(list(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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),aa(num,num,bit0,one)))))) ) ) ).

% ATP.lambda_178
tff(fact_8359_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_qe(list(A),fun(A,bool),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),aa(num,num,bit0,one)))) ) ) ) ).

% ATP.lambda_179
tff(fact_8360_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_180
tff(fact_8361_ATP_Olambda__181,axiom,
    ! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_cf(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),zero_zero(nat))) ).

% ATP.lambda_181
tff(fact_8362_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aqu(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_Pair(ref(A),product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ) ).

% ATP.lambda_182
tff(fact_8363_ATP_Olambda__183,axiom,
    ! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_bu(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ).

% ATP.lambda_183
tff(fact_8364_ATP_Olambda__184,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_aip(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),image(B,A,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_184
tff(fact_8365_ATP_Olambda__185,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_afj(fun(A,B),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_185
tff(fact_8366_ATP_Olambda__186,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_oy(set(A),fun(A,bool),Uu),Uua))
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_186
tff(fact_8367_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_arf(fun(A,A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(A,A,Uu,Uua))) ) ) ).

% ATP.lambda_187
tff(fact_8368_ATP_Olambda__188,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_sx(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_188
tff(fact_8369_ATP_Olambda__189,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_we(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_189
tff(fact_8370_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lq(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_190
tff(fact_8371_ATP_Olambda__191,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_aif(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).

% ATP.lambda_191
tff(fact_8372_ATP_Olambda__192,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_et(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).

% ATP.lambda_192
tff(fact_8373_ATP_Olambda__193,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aat(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ).

% ATP.lambda_193
tff(fact_8374_ATP_Olambda__194,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_ro(nat,fun(list(A),bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_194
tff(fact_8375_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ix(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_195
tff(fact_8376_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_iw(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_196
tff(fact_8377_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_is(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_197
tff(fact_8378_ATP_Olambda__198,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_ats(nat,fun(nat,list(nat))),Uu),Uua) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Uu),nat_list_decode(Uua)) ).

% ATP.lambda_198
tff(fact_8379_ATP_Olambda__199,axiom,
    ! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_uq(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua) = map_add(A,B,Uua,map_of(A,B,Uu)) ).

% ATP.lambda_199
tff(fact_8380_ATP_Olambda__200,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_ajq(A,fun(list(A),bool)),Uu),Uua))
    <=> pp(aa(set(A),bool,member(A,Uu),aa(list(A),set(A),set2(A),Uua))) ) ).

% ATP.lambda_200
tff(fact_8381_ATP_Olambda__201,axiom,
    ! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_xw(list(set(A)),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(set(A)),bool,member(set(A),Uua),aa(list(set(A)),set(set(A)),set2(set(A)),Uu))) ) ).

% ATP.lambda_201
tff(fact_8382_ATP_Olambda__202,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_qr(list(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,member(A,Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_202
tff(fact_8383_ATP_Olambda__203,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_apb(nat,fun(list(A),bool),Uu),Uua))
    <=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_203
tff(fact_8384_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_afw(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_204
tff(fact_8385_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_afv(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = last(A,Uu) ) ) ).

% ATP.lambda_205
tff(fact_8386_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_pl(nat,fun(nat,bool)),Uu),Uua))
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_206
tff(fact_8387_ATP_Olambda__207,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ls(set(A),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ).

% ATP.lambda_207
tff(fact_8388_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ck(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_208
tff(fact_8389_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bf(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_209
tff(fact_8390_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ak(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_210
tff(fact_8391_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bh(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_211
tff(fact_8392_ATP_Olambda__212,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aup(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_212
tff(fact_8393_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mo(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_213
tff(fact_8394_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_gm(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_214
tff(fact_8395_ATP_Olambda__215,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aah(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_215
tff(fact_8396_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vh(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_216
tff(fact_8397_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahg(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_217
tff(fact_8398_ATP_Olambda__218,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_218
tff(fact_8399_ATP_Olambda__219,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bg(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_219
tff(fact_8400_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_apv(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_220
tff(fact_8401_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_al(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_221
tff(fact_8402_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bi(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_222
tff(fact_8403_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ax(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_223
tff(fact_8404_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_mp(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_224
tff(fact_8405_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_jm(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_225
tff(fact_8406_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vw(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_226
tff(fact_8407_ATP_Olambda__227,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_wj(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_227
tff(fact_8408_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ve(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_228
tff(fact_8409_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vo(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_229
tff(fact_8410_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu: A,Uua: multiset(A)] : aa(multiset(A),nat,aTP_Lamp_akn(A,fun(multiset(A),nat),Uu),Uua) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uu) ).

% ATP.lambda_230
tff(fact_8411_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_atv(multiset(A),fun(multiset(A),bool)),Uu),Uua))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Uua),Uu)) ) ).

% ATP.lambda_231
tff(fact_8412_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ys(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_232
tff(fact_8413_ATP_Olambda__233,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_xt(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_233
tff(fact_8414_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_zo(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_234
tff(fact_8415_ATP_Olambda__235,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_wi(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_235
tff(fact_8416_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_236
tff(fact_8417_ATP_Olambda__237,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_wh(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_237
tff(fact_8418_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahj(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_238
tff(fact_8419_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vd(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_239
tff(fact_8420_ATP_Olambda__240,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_atw(multiset(A),fun(multiset(A),bool)),Uu),Uua))
    <=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Uua),Uu)) ) ).

% ATP.lambda_240
tff(fact_8421_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: ref(A)] : aa(ref(A),assn,aa(A,fun(ref(A),assn),aTP_Lamp_ca(A,fun(ref(A),assn)),Uu),Uua) = sngr_assn(A,Uua,Uu) ) ).

% ATP.lambda_241
tff(fact_8422_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aa(list(A),fun(array(A),assn),aTP_Lamp_cb(list(A),fun(array(A),assn)),Uu),Uua) = snga_assn(A,Uua,Uu) ) ).

% ATP.lambda_242
tff(fact_8423_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_lh(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).

% ATP.lambda_243
tff(fact_8424_ATP_Olambda__244,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_lx(int,fun(int,bool),Uu),Uua))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uua),Uu)) ) ).

% ATP.lambda_244
tff(fact_8425_ATP_Olambda__245,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ds(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),Uu)) ) ) ).

% ATP.lambda_245
tff(fact_8426_ATP_Olambda__246,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_abs(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).

% ATP.lambda_246
tff(fact_8427_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_np(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).

% ATP.lambda_247
tff(fact_8428_ATP_Olambda__248,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_oz(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_248
tff(fact_8429_ATP_Olambda__249,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ta(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_249
tff(fact_8430_ATP_Olambda__250,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_pc(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_250
tff(fact_8431_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_251
tff(fact_8432_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),nat,aTP_Lamp_aoq(array(A),fun(heap_ext(product_unit),nat),Uu),Uua) = array_length(A,Uua,Uu) ) ).

% ATP.lambda_252
tff(fact_8433_ATP_Olambda__253,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_so(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).

% ATP.lambda_253
tff(fact_8434_ATP_Olambda__254,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),A,aTP_Lamp_bz(ref(A),fun(heap_ext(product_unit),A),Uu),Uua) = ref_get(A,Uua,Uu) ) ).

% ATP.lambda_254
tff(fact_8435_ATP_Olambda__255,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aju(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,Uua),Uu)) ) ).

% ATP.lambda_255
tff(fact_8436_ATP_Olambda__256,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_be(set(B),fun(B,bool),Uu),Uua))
    <=> pp(aa(set(B),bool,member(B,Uua),Uu)) ) ).

% ATP.lambda_256
tff(fact_8437_ATP_Olambda__257,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ajt(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ) ).

% ATP.lambda_257
tff(fact_8438_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ajy(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ) ).

% ATP.lambda_258
tff(fact_8439_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),Uu),Uua))
    <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ).

% ATP.lambda_259
tff(fact_8440_ATP_Olambda__260,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Uua),Uu) ).

% ATP.lambda_260
tff(fact_8441_ATP_Olambda__261,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_aan(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_261
tff(fact_8442_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_rn(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_262
tff(fact_8443_ATP_Olambda__263,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(C,set(B))] : aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_ago(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua) = aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,Uua),Uu) ).

% ATP.lambda_263
tff(fact_8444_ATP_Olambda__264,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ap(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_264
tff(fact_8445_ATP_Olambda__265,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_sc(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_265
tff(fact_8446_ATP_Olambda__266,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_sd(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Uua),nil(list(A))) ).

% ATP.lambda_266
tff(fact_8447_ATP_Olambda__267,axiom,
    ! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_aeb(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_267
tff(fact_8448_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_aqv(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aqu(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,Uu,Uua)) ) ).

% ATP.lambda_268
tff(fact_8449_ATP_Olambda__269,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aki(multiset(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uua))) ) ).

% ATP.lambda_269
tff(fact_8450_ATP_Olambda__270,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_aoy(fun(B,nat),fun(B,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(B,nat,Uu,Uua))) ) ).

% ATP.lambda_270
tff(fact_8451_ATP_Olambda__271,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_kx(fun(A,nat),fun(A,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua))) ) ).

% ATP.lambda_271
tff(fact_8452_ATP_Olambda__272,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aks(set(multiset(A)),fun(A,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_akn(A,fun(multiset(A),nat),Uua)),Uu)))) ) ).

% ATP.lambda_272
tff(fact_8453_ATP_Olambda__273,axiom,
    ! [A: $tType,Uu: code_natural,Uua: A] :
      ( pp(aa(A,bool,aa(code_natural,fun(A,bool),aTP_Lamp_qo(code_natural,fun(A,bool)),Uu),Uua))
    <=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),Uu)) ) ).

% ATP.lambda_273
tff(fact_8454_ATP_Olambda__274,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aso(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_274
tff(fact_8455_ATP_Olambda__275,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_aca(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_275
tff(fact_8456_ATP_Olambda__276,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_sj(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).

% ATP.lambda_276
tff(fact_8457_ATP_Olambda__277,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_sk(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_277
tff(fact_8458_ATP_Olambda__278,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_nj(fun(B,C),fun(product_prod(A,B),C),Uu),Uua) = aa(B,C,Uu,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).

% ATP.lambda_278
tff(fact_8459_ATP_Olambda__279,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_os(fun(A,C),fun(product_prod(A,B),C),Uu),Uua) = aa(A,C,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_279
tff(fact_8460_ATP_Olambda__280,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ajv(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_280
tff(fact_8461_ATP_Olambda__281,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_if(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_281
tff(fact_8462_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_282
tff(fact_8463_ATP_Olambda__283,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_sz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_283
tff(fact_8464_ATP_Olambda__284,axiom,
    ! [A: $tType,C: $tType,Uu: C,Uua: fun(C,set(set(A)))] : aa(fun(C,set(set(A))),set(set(A)),aTP_Lamp_asa(C,fun(fun(C,set(set(A))),set(set(A))),Uu),Uua) = aa(C,set(set(A)),Uua,Uu) ).

% ATP.lambda_284
tff(fact_8465_ATP_Olambda__285,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,set(A))] : aa(fun(B,set(A)),set(A),aTP_Lamp_agn(B,fun(fun(B,set(A)),set(A)),Uu),Uua) = aa(B,set(A),Uua,Uu) ).

% ATP.lambda_285
tff(fact_8466_ATP_Olambda__286,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_akm(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).

% ATP.lambda_286
tff(fact_8467_ATP_Olambda__287,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_wm(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_287
tff(fact_8468_ATP_Olambda__288,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_yi(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_288
tff(fact_8469_ATP_Olambda__289,axiom,
    ! [A: $tType,Uu: fun(product_unit,A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_cu(fun(product_unit,A),fun(product_unit,A),Uu),Uua) = aa(product_unit,A,Uu,product_Unity) ).

% ATP.lambda_289
tff(fact_8470_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aha(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_min(A),Uu),Uua)) ) ).

% ATP.lambda_290
tff(fact_8471_ATP_Olambda__291,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ags(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_291
tff(fact_8472_ATP_Olambda__292,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_agm(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_292
tff(fact_8473_ATP_Olambda__293,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_agr(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_293
tff(fact_8474_ATP_Olambda__294,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: option(multiset(A))] : aa(option(multiset(A)),option(multiset(A)),aa(multiset(A),fun(option(multiset(A)),option(multiset(A))),aTP_Lamp_aug(multiset(A),fun(option(multiset(A)),option(multiset(A)))),Uu),Uua) = aa(multiset(A),option(multiset(A)),some(multiset(A)),case_option(multiset(A),multiset(A),Uu,aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uu),Uua)) ).

% ATP.lambda_294
tff(fact_8475_ATP_Olambda__295,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: option(multiset(A))] : aa(option(multiset(A)),option(multiset(A)),aa(multiset(A),fun(option(multiset(A)),option(multiset(A))),aTP_Lamp_aud(multiset(A),fun(option(multiset(A)),option(multiset(A)))),Uu),Uua) = aa(multiset(A),option(multiset(A)),some(multiset(A)),case_option(multiset(A),multiset(A),Uu,aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uu),Uua)) ).

% ATP.lambda_295
tff(fact_8476_ATP_Olambda__296,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_fe(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_296
tff(fact_8477_ATP_Olambda__297,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_afl(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_afk(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_297
tff(fact_8478_ATP_Olambda__298,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),list(B)),aTP_Lamp_pv(fun(B,A),fun(list(B),fun(product_prod(list(B),list(B)),list(B))),Uu),Uua) = aa(fun(list(B),fun(list(B),list(B))),fun(product_prod(list(B),list(B)),list(B)),product_case_prod(list(B),list(B),list(B)),aa(list(B),fun(list(B),fun(list(B),list(B))),aTP_Lamp_pu(fun(B,A),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua)) ) ).

% ATP.lambda_298
tff(fact_8479_ATP_Olambda__299,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(bool)),aTP_Lamp_ps(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Uu),Uua) = aa(fun(A,fun(list(A),option(bool))),fun(product_prod(A,list(A)),option(bool)),product_case_prod(A,list(A),option(bool)),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_pr(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua)) ).

% ATP.lambda_299
tff(fact_8480_ATP_Olambda__300,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_pp(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_po(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).

% ATP.lambda_300
tff(fact_8481_ATP_Olambda__301,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_pi(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_ph(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_301
tff(fact_8482_ATP_Olambda__302,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ny(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = 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)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_nx(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_302
tff(fact_8483_ATP_Olambda__303,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = 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)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_nv(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_303
tff(fact_8484_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_nu(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_nt(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_304
tff(fact_8485_ATP_Olambda__305,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ns(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_nr(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_305
tff(fact_8486_ATP_Olambda__306,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_nn(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = 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)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_nm(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_306
tff(fact_8487_ATP_Olambda__307,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ei(fun(A,heap_Time_Heap(B)),fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua) = aa(fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(heap_ext(product_unit),nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_eh(fun(A,heap_Time_Heap(B)),fun(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).

% ATP.lambda_307
tff(fact_8488_ATP_Olambda__308,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ef(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua) = aa(fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),product_case_prod(heap_ext(product_unit),nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_ee(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).

% ATP.lambda_308
tff(fact_8489_ATP_Olambda__309,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,bool)] : aa(fun(A,bool),set(A),aTP_Lamp_api(fun(A,fun(B,bool)),fun(fun(A,bool),set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aph(fun(A,fun(B,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua)) ).

% ATP.lambda_309
tff(fact_8490_ATP_Olambda__310,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] : aa(B,set(A),aTP_Lamp_ads(fun(A,fun(B,bool)),fun(B,set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_adr(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua)) ).

% ATP.lambda_310
tff(fact_8491_ATP_Olambda__311,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_aae(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),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_aab(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_311
tff(fact_8492_ATP_Olambda__312,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_aaa(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_zz(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_312
tff(fact_8493_ATP_Olambda__313,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_aac(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_aab(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_313
tff(fact_8494_ATP_Olambda__314,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_aad(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_zz(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_314
tff(fact_8495_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu: fun(heap_ext(product_unit),A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_bq(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,Uu,Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat)))) ).

% ATP.lambda_315
tff(fact_8496_ATP_Olambda__316,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_su(fun(A,option(B)),fun(A,bool),Uu),Uua))
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_316
tff(fact_8497_ATP_Olambda__317,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_acf(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),image2(B,set(product_prod(A,B)),aTP_Lamp_ace(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_317
tff(fact_8498_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: A] : aa(A,nat,aTP_Lamp_akv(set(multiset(A)),fun(A,nat),Uu),Uua) = aa(set(nat),nat,complete_Sup_Sup(nat),aa(set(multiset(A)),set(nat),image2(multiset(A),nat,aTP_Lamp_akn(A,fun(multiset(A),nat),Uua)),Uu)) ).

% ATP.lambda_318
tff(fact_8499_ATP_Olambda__319,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_ua(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_tz(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).

% ATP.lambda_319
tff(fact_8500_ATP_Olambda__320,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ex(nat,fun(nat,bool),Uu),Uua))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))),Uua)))) ) ).

% ATP.lambda_320
tff(fact_8501_ATP_Olambda__321,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_yn(set(set(A)),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Sup_Sup(bool),aa(set(set(A)),set(bool),image2(set(A),bool,member(A,Uua)),Uu))) ) ).

% ATP.lambda_321
tff(fact_8502_ATP_Olambda__322,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_zx(set(set(A)),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Inf_Inf(bool),aa(set(set(A)),set(bool),image2(set(A),bool,member(A,Uua)),Uu))) ) ).

% ATP.lambda_322
tff(fact_8503_ATP_Olambda__323,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_ey(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_323
tff(fact_8504_ATP_Olambda__324,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_aag(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)),image2(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_324
tff(fact_8505_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dj(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_325
tff(fact_8506_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_db(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_326
tff(fact_8507_ATP_Olambda__327,axiom,
    ! [A: $tType,Uu: list(A),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_apn(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(nat,A,nth(A,Uu),aa(code_natural,nat,code_nat_of_natural,Uua))) ).

% ATP.lambda_327
tff(fact_8508_ATP_Olambda__328,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_qq(list(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,member(A,Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_328
tff(fact_8509_ATP_Olambda__329,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_abz(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_329
tff(fact_8510_ATP_Olambda__330,axiom,
    ! [P7: $tType,O: $tType,Uu: fun(O,set(P7)),Uua: set(O)] : aa(set(O),set(P7),aTP_Lamp_xu(fun(O,set(P7)),fun(set(O),set(P7)),Uu),Uua) = aa(set(set(P7)),set(P7),complete_Sup_Sup(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),Uu),Uua)) ).

% ATP.lambda_330
tff(fact_8511_ATP_Olambda__331,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_amu(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,Uu),Uua)) ) ).

% ATP.lambda_331
tff(fact_8512_ATP_Olambda__332,axiom,
    ! [D: $tType,B: $tType,Uu: set(B),Uua: fun(B,set(D))] : aa(fun(B,set(D)),set(D),aa(set(B),fun(fun(B,set(D)),set(D)),aTP_Lamp_anu(set(B),fun(fun(B,set(D)),set(D))),Uu),Uua) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),image2(B,set(D),Uua),Uu)) ).

% ATP.lambda_332
tff(fact_8513_ATP_Olambda__333,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_anr(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),Uu)) ) ).

% ATP.lambda_333
tff(fact_8514_ATP_Olambda__334,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(set(A),fun(fun(A,bool),bool),aTP_Lamp_aln(set(A),fun(fun(A,bool),bool)),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Sup_Sup(bool),aa(set(A),set(bool),image2(A,bool,Uua),Uu))) ) ).

% ATP.lambda_334
tff(fact_8515_ATP_Olambda__335,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: fun(A,set(C))] : aa(fun(A,set(C)),set(C),aa(set(A),fun(fun(A,set(C)),set(C)),aTP_Lamp_ant(set(A),fun(fun(A,set(C)),set(C))),Uu),Uua) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),Uua),Uu)) ).

% ATP.lambda_335
tff(fact_8516_ATP_Olambda__336,axiom,
    ! [C: $tType,A: $tType] :
      ( complete_Sup(C)
     => ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_anq(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,Uua),Uu)) ) ).

% ATP.lambda_336
tff(fact_8517_ATP_Olambda__337,axiom,
    ! [P7: $tType,O: $tType,Uu: fun(O,set(P7)),Uua: set(O)] : aa(set(O),set(P7),aTP_Lamp_zl(fun(O,set(P7)),fun(set(O),set(P7)),Uu),Uua) = aa(set(set(P7)),set(P7),complete_Inf_Inf(set(P7)),aa(set(O),set(set(P7)),image2(O,set(P7),Uu),Uua)) ).

% ATP.lambda_337
tff(fact_8518_ATP_Olambda__338,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_amv(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,Uu),Uua)) ) ).

% ATP.lambda_338
tff(fact_8519_ATP_Olambda__339,axiom,
    ! [C: $tType,B: $tType] :
      ( complete_Inf(C)
     => ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_anp(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),Uu)) ) ).

% ATP.lambda_339
tff(fact_8520_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(set(A),fun(fun(A,bool),bool),aTP_Lamp_ama(set(A),fun(fun(A,bool),bool)),Uu),Uua))
    <=> pp(aa(set(bool),bool,complete_Inf_Inf(bool),aa(set(A),set(bool),image2(A,bool,Uua),Uu))) ) ).

% ATP.lambda_340
tff(fact_8521_ATP_Olambda__341,axiom,
    ! [C: $tType,A: $tType] :
      ( complete_Inf(C)
     => ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_ano(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,Uua),Uu)) ) ).

% ATP.lambda_341
tff(fact_8522_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( sup(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_am(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_342
tff(fact_8523_ATP_Olambda__343,axiom,
    ! [A: $tType] :
      ( inf(A)
     => ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_cr(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_343
tff(fact_8524_ATP_Olambda__344,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_fa(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),modulo_modulo(code_natural,Uua,Uu)) ).

% ATP.lambda_344
tff(fact_8525_ATP_Olambda__345,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_atc(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_345
tff(fact_8526_ATP_Olambda__346,axiom,
    ! [A: $tType,Uu: list(product_prod(code_natural,A)),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_ty(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(code_natural,A,pick(A,Uu),Uua)) ).

% ATP.lambda_346
tff(fact_8527_ATP_Olambda__347,axiom,
    ! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: A] : aa(A,B,aTP_Lamp_wf(list(product_prod(A,B)),fun(A,B),Uu),Uua) = aa(option(B),B,the2(B),aa(A,option(B),map_of(A,B,Uu),Uua)) ).

% ATP.lambda_347
tff(fact_8528_ATP_Olambda__348,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_vb(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = aa(product_prod(B,A),fun(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)) ).

% ATP.lambda_348
tff(fact_8529_ATP_Olambda__349,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_acg(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = aa(product_prod(A,B),fun(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),Uu),Uua)) ).

% ATP.lambda_349
tff(fact_8530_ATP_Olambda__350,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aeu(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_350
tff(fact_8531_ATP_Olambda__351,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aev(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_351
tff(fact_8532_ATP_Olambda__352,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ml(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_352
tff(fact_8533_ATP_Olambda__353,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mk(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_353
tff(fact_8534_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_rg(A,fun(A,bool),Uu),Uua))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uua)) ) ) ).

% ATP.lambda_354
tff(fact_8535_ATP_Olambda__355,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_dh(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,member(A,Uua),Uu)) ) ).

% ATP.lambda_355
tff(fact_8536_ATP_Olambda__356,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_amf(A,fun(A,bool)),Uu),Uua))
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_356
tff(fact_8537_ATP_Olambda__357,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aps(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_357
tff(fact_8538_ATP_Olambda__358,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_apw(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_358
tff(fact_8539_ATP_Olambda__359,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_359
tff(fact_8540_ATP_Olambda__360,axiom,
    ! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_aec(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A)))) ).

% ATP.lambda_360
tff(fact_8541_ATP_Olambda__361,axiom,
    ! [Uu: nat,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_cv(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),Uu))) ).

% ATP.lambda_361
tff(fact_8542_ATP_Olambda__362,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_aga(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_362
tff(fact_8543_ATP_Olambda__363,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_arg(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_363
tff(fact_8544_ATP_Olambda__364,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,set(product_prod(C,C)),aTP_Lamp_asj(fun(B,set(C)),fun(B,set(product_prod(C,C))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Uu,Uua)) ).

% ATP.lambda_364
tff(fact_8545_ATP_Olambda__365,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_ask(fun(A,set(B)),fun(A,set(product_prod(B,B))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_365
tff(fact_8546_ATP_Olambda__366,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_me(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_366
tff(fact_8547_ATP_Olambda__367,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [Uu: fun(B,rat),Uua: B] : aa(B,A,aTP_Lamp_aer(fun(B,rat),fun(B,A),Uu),Uua) = aa(rat,A,field_char_0_of_rat(A),aa(B,rat,Uu,Uua)) ) ).

% ATP.lambda_367
tff(fact_8548_ATP_Olambda__368,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_fv(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).

% ATP.lambda_368
tff(fact_8549_ATP_Olambda__369,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_ic(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_369
tff(fact_8550_ATP_Olambda__370,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_fm(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_370
tff(fact_8551_ATP_Olambda__371,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_yk(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_371
tff(fact_8552_ATP_Olambda__372,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_zg(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_372
tff(fact_8553_ATP_Olambda__373,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_gf(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_373
tff(fact_8554_ATP_Olambda__374,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_dm(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),Uu),Uua) = transitive_trancl(B,aa(A,set(product_prod(B,B)),Uu,Uua)) ).

% ATP.lambda_374
tff(fact_8555_ATP_Olambda__375,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_jc(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_375
tff(fact_8556_ATP_Olambda__376,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_id(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_376
tff(fact_8557_ATP_Olambda__377,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_fl(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_377
tff(fact_8558_ATP_Olambda__378,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_bv(fun(B,A),fun(B,heap_Time_Heap(A)),Uu),Uua) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_378
tff(fact_8559_ATP_Olambda__379,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gw(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_379
tff(fact_8560_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_im(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_380
tff(fact_8561_ATP_Olambda__381,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_fs(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_381
tff(fact_8562_ATP_Olambda__382,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_yo(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_382
tff(fact_8563_ATP_Olambda__383,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_mr(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_383
tff(fact_8564_ATP_Olambda__384,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_xf(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_384
tff(fact_8565_ATP_Olambda__385,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_aib(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_385
tff(fact_8566_ATP_Olambda__386,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: A] : aa(A,assn,aTP_Lamp_bl(fun(A,fun(B,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),Uu,Uua)) ).

% ATP.lambda_386
tff(fact_8567_ATP_Olambda__387,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: C] : aa(C,set(product_prod(A,B)),aTP_Lamp_akc(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),Uu),Uua) = converse(B,A,aa(C,set(product_prod(B,A)),Uu,Uua)) ).

% ATP.lambda_387
tff(fact_8568_ATP_Olambda__388,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_rx(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_388
tff(fact_8569_ATP_Olambda__389,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_ahf(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).

% ATP.lambda_389
tff(fact_8570_ATP_Olambda__390,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_anz(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_390
tff(fact_8571_ATP_Olambda__391,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_aoe(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = principal(D,aa(C,set(D),Uu,Uua)) ).

% ATP.lambda_391
tff(fact_8572_ATP_Olambda__392,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_agi(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = principal(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_392
tff(fact_8573_ATP_Olambda__393,axiom,
    ! [E: $tType,A: $tType,Uu: fun(A,set(E)),Uua: A] : aa(A,filter(E),aTP_Lamp_aod(fun(A,set(E)),fun(A,filter(E)),Uu),Uua) = principal(E,aa(A,set(E),Uu,Uua)) ).

% ATP.lambda_393
tff(fact_8574_ATP_Olambda__394,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_agh(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_394
tff(fact_8575_ATP_Olambda__395,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_yb(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_395
tff(fact_8576_ATP_Olambda__396,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(B),aTP_Lamp_asl(fun(A,set(product_prod(B,B))),fun(A,set(B)),Uu),Uua) = field2(B,aa(A,set(product_prod(B,B)),Uu,Uua)) ).

% ATP.lambda_396
tff(fact_8577_ATP_Olambda__397,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_aey(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_397
tff(fact_8578_ATP_Olambda__398,axiom,
    ! [A: $tType,Uu: fun(set(A),fun(A,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_are(fun(set(A),fun(A,bool)),fun(set(A),set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),Uu,Uua)) ).

% ATP.lambda_398
tff(fact_8579_ATP_Olambda__399,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: A] : aa(A,set(B),aTP_Lamp_abi(fun(A,fun(B,bool)),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),Uu,Uua)) ).

% ATP.lambda_399
tff(fact_8580_ATP_Olambda__400,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_vv(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_400
tff(fact_8581_ATP_Olambda__401,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_aak(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_401
tff(fact_8582_ATP_Olambda__402,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_mq(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_402
tff(fact_8583_ATP_Olambda__403,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_vg(fun(B,bool),fun(B,bool),Uu),Uua))
    <=> ~ pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_403
tff(fact_8584_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_dg(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_404
tff(fact_8585_ATP_Olambda__405,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_aqy(fun(B,fun(A,bool)),fun(B,bool),Uu),Uua))
    <=> ! [X_12: A] : pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uua),X_12)) ) ).

% ATP.lambda_405
tff(fact_8586_ATP_Olambda__406,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(A,fun(B,bool)),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_apu(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
        <=> ! [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_12)) ) ) ).

% ATP.lambda_406
tff(fact_8587_ATP_Olambda__407,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_alf(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ! [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_12)) ) ).

% ATP.lambda_407
tff(fact_8588_ATP_Olambda__408,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adq(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_12)) ) ).

% ATP.lambda_408
tff(fact_8589_ATP_Olambda__409,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_asp(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cl(nat,fun(nat,bool)),Uu)) ).

% ATP.lambda_409
tff(fact_8590_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_aoi(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),bool),aTP_Lamp_aoh(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua))) ).

% ATP.lambda_410
tff(fact_8591_ATP_Olambda__411,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool)] : aa(fun(B,bool),filter(product_prod(A,B)),aa(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B))),aTP_Lamp_atf(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))),Uu),Uua) = principal(product_prod(A,B),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_aax(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua)))) ).

% ATP.lambda_411
tff(fact_8592_ATP_Olambda__412,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_asv(fun(A,fun(B,bool)),fun(B,bool),Uu),Uua))
    <=> ? [A9: A] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,A9),Uua)) ) ).

% ATP.lambda_412
tff(fact_8593_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adx(list(A),fun(A,bool),Uu),Uua))
    <=> ? [I: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).

% ATP.lambda_413
tff(fact_8594_ATP_Olambda__414,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(option(A))),Uua: set(option(A))] :
          ( pp(aa(set(option(A)),bool,aTP_Lamp_amy(set(set(option(A))),fun(set(option(A)),bool),Uu),Uua))
        <=> ? [F11: fun(set(option(A)),option(A))] :
              ( ( Uua = aa(set(set(option(A))),set(option(A)),image2(set(option(A)),option(A),F11),Uu) )
              & ! [X4: set(option(A))] :
                  ( pp(aa(set(set(option(A))),bool,member(set(option(A)),X4),Uu))
                 => pp(aa(set(option(A)),bool,member(option(A),aa(set(option(A)),option(A),F11,X4)),X4)) ) ) ) ) ).

% ATP.lambda_414
tff(fact_8595_ATP_Olambda__415,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( pp(aa(set(B),bool,aTP_Lamp_amw(set(set(B)),fun(set(B),bool),Uu),Uua))
    <=> ? [F11: fun(set(B),B)] :
          ( ( Uua = aa(set(set(B)),set(B),image2(set(B),B,F11),Uu) )
          & ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X4),Uu))
             => pp(aa(set(B),bool,member(B,aa(set(B),B,F11,X4)),X4)) ) ) ) ).

% ATP.lambda_415
tff(fact_8596_ATP_Olambda__416,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_amp(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F11: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F11),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F11,X4)),X4)) ) ) ) ) ).

% ATP.lambda_416
tff(fact_8597_ATP_Olambda__417,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_amt(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F11: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F11),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F11,X4)),X4)) ) ) ) ) ).

% ATP.lambda_417
tff(fact_8598_ATP_Olambda__418,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_ams(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F11: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F11),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F11,X4)),X4)) ) ) ) ) ).

% ATP.lambda_418
tff(fact_8599_ATP_Olambda__419,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_adk(set(A),fun(set(A),bool),Uu),Uua))
    <=> ? [B11: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B11) )
          & pp(aa(set(set(A)),bool,member(set(A),Uu),pow(A,B11))) ) ) ).

% ATP.lambda_419
tff(fact_8600_ATP_Olambda__420,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( pp(aa(filter(A),bool,aTP_Lamp_amr(set(filter(A)),fun(filter(A),bool),Uu),Uua))
    <=> ! [X4: filter(A)] :
          ( pp(aa(set(filter(A)),bool,member(filter(A),X4),Uu))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),Uua),X4)) ) ) ).

% ATP.lambda_420
tff(fact_8601_ATP_Olambda__421,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_asu(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X4)) ) ) ) ).

% ATP.lambda_421
tff(fact_8602_ATP_Olambda__422,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_amj(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X4)) ) ) ) ).

% ATP.lambda_422
tff(fact_8603_ATP_Olambda__423,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_arr(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Uua)) ) ) ) ).

% ATP.lambda_423
tff(fact_8604_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_amk(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Uua)) ) ) ) ).

% ATP.lambda_424
tff(fact_8605_ATP_Olambda__425,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_anb(set(multiset(A)),fun(multiset(A),bool),Uu),Uua))
    <=> ! [X4: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),Uu))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Uua),X4)) ) ) ).

% ATP.lambda_425
tff(fact_8606_ATP_Olambda__426,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A)] :
      ( pp(aa(multiset(A),bool,aTP_Lamp_ana(set(multiset(A)),fun(multiset(A),bool),Uu),Uua))
    <=> ! [X4: multiset(A)] :
          ( pp(aa(set(multiset(A)),bool,member(multiset(A),X4),Uu))
         => pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X4),Uua)) ) ) ).

% ATP.lambda_426
tff(fact_8607_ATP_Olambda__427,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aTP_Lamp_aqo(set(filter(A)),fun(fun(A,bool),bool),Uu),Uua))
    <=> ! [X4: filter(A)] :
          ( pp(aa(set(filter(A)),bool,member(filter(A),X4),Uu))
         => eventually(A,Uua,X4) ) ) ).

% ATP.lambda_427
tff(fact_8608_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_amd(set(set(A)),fun(A,bool),Uu),Uua))
    <=> ! [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
         => pp(aa(set(A),bool,member(A,Uua),X4)) ) ) ).

% ATP.lambda_428
tff(fact_8609_ATP_Olambda__429,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_alq(set(set(A)),fun(A,bool),Uu),Uua))
    <=> ? [X4: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X4),Uu))
          & pp(aa(set(A),bool,member(A,Uua),X4)) ) ) ).

% ATP.lambda_429
tff(fact_8610_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aqe(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y4))
             => pp(aa(A,bool,Uu,Y4)) ) ) ) ).

% ATP.lambda_430
tff(fact_8611_ATP_Olambda__431,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ans(fun(A,bool),fun(set(A),bool),Uu),Uua))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
         => pp(aa(A,bool,Uu,X4)) ) ) ).

% ATP.lambda_431
tff(fact_8612_ATP_Olambda__432,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_auj(set(product_prod(A,A)),fun(set(A),bool),Uu),Uua))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),Uua))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),Uu))
                | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_432
tff(fact_8613_ATP_Olambda__433,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_adj(fun(A,option(B)),fun(B,bool),Uu),Uua))
    <=> ? [A9: A] : aa(A,option(B),Uu,A9) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_433
tff(fact_8614_ATP_Olambda__434,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_adw(fun(A,assn),fun(product_prod(heap_ext(product_unit),set(nat)),bool),Uu),Uua))
    <=> ? [X4: A] : pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(A,assn,Uu,X4)),Uua)) ) ).

% ATP.lambda_434
tff(fact_8615_ATP_Olambda__435,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_acs(A,fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [V7: B] : Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),V7) ) ).

% ATP.lambda_435
tff(fact_8616_ATP_Olambda__436,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adg(fun(B,A),fun(A,bool),Uu),Uua))
    <=> ? [X4: B] : Uua = aa(B,A,Uu,X4) ) ).

% ATP.lambda_436
tff(fact_8617_ATP_Olambda__437,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,bool),Uua: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aTP_Lamp_ani(fun(B,bool),fun(fun(A,B),bool),Uu),Uua))
    <=> ! [X4: A] : pp(aa(B,bool,Uu,aa(A,B,Uua,X4))) ) ).

% ATP.lambda_437
tff(fact_8618_ATP_Olambda__438,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: product_prod(product_prod(bool,A),product_prod(bool,B))] :
      ( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_aed(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu),Uua))
    <=> ? [X4: A,Y4: B] :
          ( ( Uua = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fFalse),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y4)) )
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4)),Uu)) ) ) ).

% ATP.lambda_438
tff(fact_8619_ATP_Olambda__439,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_adu(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [A9: A,B7: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B7) )
          & ( aa(A,option(B),Uu,A9) = aa(B,option(B),some(B),B7) ) ) ) ).

% ATP.lambda_439
tff(fact_8620_ATP_Olambda__440,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( pp(aa(product_prod(set(A),set(A)),bool,aTP_Lamp_alz(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
    <=> ? [X14: set(A),Y9: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X14),Y9) )
          & ( X14 != bot_bot(set(A)) )
          & ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),Y9))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,member(A,Xa3),X14))
                  & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_440
tff(fact_8621_ATP_Olambda__441,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_abc(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_441
tff(fact_8622_ATP_Olambda__442,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_afb(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ).

% ATP.lambda_442
tff(fact_8623_ATP_Olambda__443,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: array(A)] : aa(array(A),heap_Time_Heap(array(A)),aTP_Lamp_no(array(A),fun(array(A),heap_Time_Heap(array(A))),Uu),Uua) = aa(array(A),heap_Time_Heap(array(A)),heap_Time_ureturn(array(A)),Uu) ) ).

% ATP.lambda_443
tff(fact_8624_ATP_Olambda__444,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_ud(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_444
tff(fact_8625_ATP_Olambda__445,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: C] : aa(C,set(B),aTP_Lamp_asg(set(product_prod(B,B)),fun(C,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_445
tff(fact_8626_ATP_Olambda__446,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_asi(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_446
tff(fact_8627_ATP_Olambda__447,axiom,
    ! [C: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: C] : aa(C,set(A),aTP_Lamp_asf(set(product_prod(A,A)),fun(C,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_447
tff(fact_8628_ATP_Olambda__448,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_ash(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_448
tff(fact_8629_ATP_Olambda__449,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_aiy(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_449
tff(fact_8630_ATP_Olambda__450,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_abd(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_450
tff(fact_8631_ATP_Olambda__451,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,set(B),aTP_Lamp_aay(fun(B,bool),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),Uu) ).

% ATP.lambda_451
tff(fact_8632_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(set(A),set(A)),aa(A,fun(list(A),fun(set(A),set(A))),aTP_Lamp_aog(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert(A),Uu) ).

% ATP.lambda_452
tff(fact_8633_ATP_Olambda__453,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_aox(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_453
tff(fact_8634_ATP_Olambda__454,axiom,
    ! [A: $tType,Uu: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_pm(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).

% ATP.lambda_454
tff(fact_8635_ATP_Olambda__455,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_fc(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),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),aa(num,num,bit0,one))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(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),aa(num,num,bit0,one))),Uua)),Uub)) ).

% ATP.lambda_455
tff(fact_8636_ATP_Olambda__456,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_du(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),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),aa(num,num,bit0,one))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(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),aa(num,num,bit0,one))),Uua)),Uub)) ).

% ATP.lambda_456
tff(fact_8637_ATP_Olambda__457,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_dv(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),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),aa(num,num,bit0,one))),Uua)),one_one(int))),aa(int,int,aa(int,fun(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),aa(num,num,bit0,one))),Uua)),Uub)) ).

% ATP.lambda_457
tff(fact_8638_ATP_Olambda__458,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_dw(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),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),aa(num,num,bit0,one))),Uua)),one_one(A))),aa(A,A,aa(A,fun(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),aa(num,num,bit0,one))),Uua)),Uub)) ) ).

% ATP.lambda_458
tff(fact_8639_ATP_Olambda__459,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_afs(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uu),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub))) ).

% ATP.lambda_459
tff(fact_8640_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_lc(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,member(B,Uub),Uua),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_460
tff(fact_8641_ATP_Olambda__461,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_lf(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,member(B,Uub),Uua),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_461
tff(fact_8642_ATP_Olambda__462,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kc(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_462
tff(fact_8643_ATP_Olambda__463,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kd(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_463
tff(fact_8644_ATP_Olambda__464,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kb(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_464
tff(fact_8645_ATP_Olambda__465,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ke(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_465
tff(fact_8646_ATP_Olambda__466,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(A,fun(fun(A,nat),fun(A,nat)),aTP_Lamp_akz(A,fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uu),aa(nat,nat,suc,aa(A,nat,Uua,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_466
tff(fact_8647_ATP_Olambda__467,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] : aa(A,nat,aa(A,fun(A,nat),aTP_Lamp_ank(fun(A,nat),fun(A,fun(A,nat)),Uu),Uua),Uub) = if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua),aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_467
tff(fact_8648_ATP_Olambda__468,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ako(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),Uua,zero_zero(A)) ) ).

% ATP.lambda_468
tff(fact_8649_ATP_Olambda__469,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_akp(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),Uua,zero_zero(A)) ) ).

% ATP.lambda_469
tff(fact_8650_ATP_Olambda__470,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jg(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_470
tff(fact_8651_ATP_Olambda__471,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_hz(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_471
tff(fact_8652_ATP_Olambda__472,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jf(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).

% ATP.lambda_472
tff(fact_8653_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_rh(fun(A,bool),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,Uu,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).

% ATP.lambda_473
tff(fact_8654_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_ur(fun(A,bool),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = if(set(A),aa(A,bool,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),Uub),Uub) ).

% ATP.lambda_474
tff(fact_8655_ATP_Olambda__475,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,bool),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_anj(fun(A,bool),fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = if(nat,aa(A,bool,Uu,Uub),aa(A,nat,Uua,Uub),zero_zero(nat)) ).

% ATP.lambda_475
tff(fact_8656_ATP_Olambda__476,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_kw(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_476
tff(fact_8657_ATP_Olambda__477,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_tm(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_477
tff(fact_8658_ATP_Olambda__478,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,bool),Uub: A] : aa(A,nat,aa(fun(A,bool),fun(A,nat),aTP_Lamp_anv(fun(A,nat),fun(fun(A,bool),fun(A,nat)),Uu),Uua),Uub) = if(nat,aa(A,bool,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat)) ).

% ATP.lambda_478
tff(fact_8659_ATP_Olambda__479,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_kz(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_479
tff(fact_8660_ATP_Olambda__480,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: fun(B,bool),Uub: B] : aa(B,fun(A,A),aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_afr(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),Uu),Uua),Uub) = if(fun(A,A),aa(B,bool,Uua,Uub),aa(B,fun(A,A),Uu,Uub),id(A)) ).

% ATP.lambda_480
tff(fact_8661_ATP_Olambda__481,axiom,
    ! [A: $tType,Uu: fun(heap_ext(product_unit),bool),Uua: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_au(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = if(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),bool,Uu,Uub),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Uua,Uub)),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ).

% ATP.lambda_481
tff(fact_8662_ATP_Olambda__482,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_th(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_482
tff(fact_8663_ATP_Olambda__483,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_ach(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_acg(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_483
tff(fact_8664_ATP_Olambda__484,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_vc(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_vb(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_484
tff(fact_8665_ATP_Olambda__485,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_bb(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,aa(A,option(A),some(A),Uub),aa(A,fun(A,option(A)),aTP_Lamp_ba(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub),Uua) ).

% ATP.lambda_485
tff(fact_8666_ATP_Olambda__486,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_pg(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = merges9089515139780605204_merge(A,Uu,aa(list(A),list(A),mergesort_by_rel(A,Uu),Uua),aa(list(A),list(A),mergesort_by_rel(A,Uu),Uub)) ).

% ATP.lambda_486
tff(fact_8667_ATP_Olambda__487,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_an(option(A),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = case_option(option(A),A,Uu,aTP_Lamp_am(A,fun(A,option(A)),Uub),Uua) ) ).

% ATP.lambda_487
tff(fact_8668_ATP_Olambda__488,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_uj(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_488
tff(fact_8669_ATP_Olambda__489,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aa(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_um(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub) = fun_upd(A,option(B),Uub,Uu,aa(B,option(B),some(B),Uua)) ).

% ATP.lambda_489
tff(fact_8670_ATP_Olambda__490,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ahs(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ).

% ATP.lambda_490
tff(fact_8671_ATP_Olambda__491,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: bool] : aa(bool,set(A),aa(set(A),fun(bool,set(A)),aTP_Lamp_aaj(set(A),fun(set(A),fun(bool,set(A))),Uu),Uua),Uub) = if(set(A),Uub,Uu,Uua) ).

% ATP.lambda_491
tff(fact_8672_ATP_Olambda__492,axiom,
    ! [B: $tType,A: $tType,Uu: heap_Time_Heap(A),Uua: fun(A,heap_Time_Heap(B)),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ej(heap_Time_Heap(A),fun(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = case_option(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),none(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(A,product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_ei(fun(A,heap_Time_Heap(B)),fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uua)),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Uu),Uub)) ).

% ATP.lambda_492
tff(fact_8673_ATP_Olambda__493,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: product_prod(D,B)] : aa(product_prod(D,B),C,aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_ou(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,aa(product_prod(D,B),D,product_fst(D,B),Uub))),aa(product_prod(D,B),B,product_snd(D,B),Uub)) ).

% ATP.lambda_493
tff(fact_8674_ATP_Olambda__494,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_aab(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_494
tff(fact_8675_ATP_Olambda__495,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_jp(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_495
tff(fact_8676_ATP_Olambda__496,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_gz(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_496
tff(fact_8677_ATP_Olambda__497,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_apr(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_497
tff(fact_8678_ATP_Olambda__498,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_ip(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_498
tff(fact_8679_ATP_Olambda__499,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_ja(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_499
tff(fact_8680_ATP_Olambda__500,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_zz(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_500
tff(fact_8681_ATP_Olambda__501,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_wn(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_501
tff(fact_8682_ATP_Olambda__502,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_yj(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_502
tff(fact_8683_ATP_Olambda__503,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_agg(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_503
tff(fact_8684_ATP_Olambda__504,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_wz(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_504
tff(fact_8685_ATP_Olambda__505,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_ih(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_505
tff(fact_8686_ATP_Olambda__506,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_fx(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_506
tff(fact_8687_ATP_Olambda__507,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_acu(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_507
tff(fact_8688_ATP_Olambda__508,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_sl(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_508
tff(fact_8689_ATP_Olambda__509,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(A,fun(B,bool)),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_apt(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ) ).

% ATP.lambda_509
tff(fact_8690_ATP_Olambda__510,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_bk(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_510
tff(fact_8691_ATP_Olambda__511,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_adr(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_511
tff(fact_8692_ATP_Olambda__512,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,B)),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_uz(fun(A,fun(B,B)),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(A,fun(B,B),Uu,Uub),Uua) ).

% ATP.lambda_512
tff(fact_8693_ATP_Olambda__513,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_sm(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_513
tff(fact_8694_ATP_Olambda__514,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_si(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_514
tff(fact_8695_ATP_Olambda__515,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_jk(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).

% ATP.lambda_515
tff(fact_8696_ATP_Olambda__516,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A)),Uub: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aun(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),Uu,ord_less(set(A))),Uub))
        & pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),Uua),Uub)) ) ) ).

% ATP.lambda_516
tff(fact_8697_ATP_Olambda__517,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_as(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uua),Uub))
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_517
tff(fact_8698_ATP_Olambda__518,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_iq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ip(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_518
tff(fact_8699_ATP_Olambda__519,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_jb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ja(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_519
tff(fact_8700_ATP_Olambda__520,axiom,
    ! [Uu: product_prod(code_natural,code_natural),Uua: code_natural,Uub: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_nf(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_ne(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uua),Uub)),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,code_natural),product_snd(code_natural,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,Uu))) ).

% ATP.lambda_520
tff(fact_8701_ATP_Olambda__521,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_on(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_om(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_521
tff(fact_8702_ATP_Olambda__522,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ol(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_ok(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_522
tff(fact_8703_ATP_Olambda__523,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_oj(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_oi(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_523
tff(fact_8704_ATP_Olambda__524,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_oh(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_og(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_524
tff(fact_8705_ATP_Olambda__525,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_of(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_oe(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_525
tff(fact_8706_ATP_Olambda__526,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ob(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_oa(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_526
tff(fact_8707_ATP_Olambda__527,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_ii(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_ih(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub),Uua) ) ).

% ATP.lambda_527
tff(fact_8708_ATP_Olambda__528,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_fy(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_fx(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub),Uua) ) ).

% ATP.lambda_528
tff(fact_8709_ATP_Olambda__529,axiom,
    ! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(B,C),Uub: product_prod(D,E)] : aa(product_prod(D,E),set(A),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_zs(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),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(D,E),fun(B,fun(C,set(A))),aTP_Lamp_zr(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uub)),Uua) ).

% ATP.lambda_529
tff(fact_8710_ATP_Olambda__530,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: filter(A),Uub: fun(B,bool)] :
      ( pp(aa(fun(B,bool),bool,aa(filter(A),fun(fun(B,bool),bool),aTP_Lamp_atb(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),Uu),Uua),Uub))
    <=> eventually(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_afe(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uub),Uua) ) ).

% ATP.lambda_530
tff(fact_8711_ATP_Olambda__531,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aop(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uub,Uu)),Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),one_one(nat))) ) ).

% ATP.lambda_531
tff(fact_8712_ATP_Olambda__532,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_jz(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,power_power(A,zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8713_ATP_Olambda__533,axiom,
    ! [A: $tType,Uu: fun(set(A),set(A)),Uua: set(A),Uub: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_arh(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),Uu),Uua),Uub) = 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)),sup_sup(set(A)),aa(set(A),set(A),Uu,Uub)),Uua)),complete_lattice_gfp(set(A),Uu)) ).

% ATP.lambda_533
tff(fact_8714_ATP_Olambda__534,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A)] : aa(set(A),set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_arc(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),Uu),Uua),Uub) = 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)),sup_sup(set(A)),aa(set(A),set(A),Uua,Uub)),Uu)),complete_lattice_gfp(set(A),Uua)) ).

% ATP.lambda_534
tff(fact_8715_ATP_Olambda__535,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_tt(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_535
tff(fact_8716_ATP_Olambda__536,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: product_prod(product_prod(A,C),product_prod(B,D))] :
      ( pp(aa(product_prod(product_prod(A,C),product_prod(B,D)),bool,aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_aus(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),basic_fsts(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),Uu))))
        & pp(aa(set(product_prod(B,D)),bool,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),bool),ord_less_eq(set(product_prod(B,D))),basic_snds(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(B,D),bool),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,bool)),fun(product_prod(B,D),bool),product_case_prod(B,D,bool),Uua)))) ) ) ).

% ATP.lambda_536
tff(fact_8717_ATP_Olambda__537,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pj(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub)))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub)),lex(A,Uu))) ) ) ) ).

% ATP.lambda_537
tff(fact_8718_ATP_Olambda__538,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_aei(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_538
tff(fact_8719_ATP_Olambda__539,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_adp(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X4: A,Y4: A,Xs6: list(A),Ys6: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4)),Uu)) ) ) ) ).

% ATP.lambda_539
tff(fact_8720_ATP_Olambda__540,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_tu(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_540
tff(fact_8721_ATP_Olambda__541,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_my(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).

% ATP.lambda_541
tff(fact_8722_ATP_Olambda__542,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_mx(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).

% ATP.lambda_542
tff(fact_8723_ATP_Olambda__543,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_aci(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_543
tff(fact_8724_ATP_Olambda__544,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ri(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).

% ATP.lambda_544
tff(fact_8725_ATP_Olambda__545,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(set(B),fun(product_prod(A,B),bool),aTP_Lamp_avb(set(A),fun(set(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),basic_fsts(A,B,Uub)),Uu))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),basic_snds(A,B,Uub)),Uua)) ) ) ).

% ATP.lambda_545
tff(fact_8726_ATP_Olambda__546,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_mu(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).

% ATP.lambda_546
tff(fact_8727_ATP_Olambda__547,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_mw(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_547
tff(fact_8728_ATP_Olambda__548,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_ts(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_548
tff(fact_8729_ATP_Olambda__549,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,option(B))] :
      ( pp(aa(fun(A,option(B)),bool,aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_aie(set(A),fun(set(B),fun(fun(A,option(B)),bool)),Uu),Uua),Uub))
    <=> ( ( dom(A,B,Uub) = Uu )
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),ran(A,B,Uub)),Uua)) ) ) ).

% ATP.lambda_549
tff(fact_8730_ATP_Olambda__550,axiom,
    ! [Uu: heap_ext(product_unit),Uua: heap_ext(product_unit),Uub: nat] :
      ( pp(aa(nat,bool,aa(heap_ext(product_unit),fun(nat,bool),aTP_Lamp_ag(heap_ext(product_unit),fun(heap_ext(product_unit),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,Uu)),Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),lim(product_unit,Uua))) ) ) ).

% ATP.lambda_550
tff(fact_8731_ATP_Olambda__551,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_mm(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,aa(nat,nat,suc,Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_551
tff(fact_8732_ATP_Olambda__552,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_td(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua)),Uu)) ) ).

% ATP.lambda_552
tff(fact_8733_ATP_Olambda__553,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_pt(set(list(A)),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A)))),Uu)) ) ).

% ATP.lambda_553
tff(fact_8734_ATP_Olambda__554,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: fun(A,A),Uua: ref(A),Uub: A] : aa(A,heap_Time_Heap(A),aa(ref(A),fun(A,heap_Time_Heap(A)),aTP_Lamp_bx(fun(A,A),fun(ref(A),fun(A,heap_Time_Heap(A))),Uu),Uua),Uub) = heap_Time_bind(product_unit,A,ref_update(A,Uua,aa(A,A,Uu,Uub)),aa(A,fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_bw(fun(A,A),fun(A,fun(product_unit,heap_Time_Heap(A))),Uu),Uub)) ) ).

% ATP.lambda_554
tff(fact_8735_ATP_Olambda__555,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_aam(nat,fun(nat,fun(set(nat),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(nat)),bool,member(set(nat),Uub),pow(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu))))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_555
tff(fact_8736_ATP_Olambda__556,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_tq(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
        & pp(aa(set(nat),bool,member(nat,Uub),Uua)) ) ) ).

% ATP.lambda_556
tff(fact_8737_ATP_Olambda__557,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_qs(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
        & pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).

% ATP.lambda_557
tff(fact_8738_ATP_Olambda__558,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: multiset(A),Uub: A] :
      ( pp(aa(A,bool,aa(multiset(A),fun(A,bool),aTP_Lamp_anw(fun(A,bool),fun(multiset(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(multiset(A),set(A),set_mset(A),Uua)))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_558
tff(fact_8739_ATP_Olambda__559,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ahb(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),aa(list(A),set(A),set2(A),Uu)))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_559
tff(fact_8740_ATP_Olambda__560,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_qj(fun(A,bool),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),aa(list(A),set(A),set2(A),Uua)))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_560
tff(fact_8741_ATP_Olambda__561,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(A,fun(product_prod(A,B),bool),aTP_Lamp_up(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Uub),graph(A,B,Uu)))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_561
tff(fact_8742_ATP_Olambda__562,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iu(A,fun(nat,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),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_562
tff(fact_8743_ATP_Olambda__563,axiom,
    ! [Uu: heap_ext(product_unit),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ad(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),lim(product_unit,Uu)))
        & ~ pp(aa(set(nat),bool,member(nat,Uub),Uua)) ) ) ).

% ATP.lambda_563
tff(fact_8744_ATP_Olambda__564,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_anc(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu)) ) ) ) ).

% ATP.lambda_564
tff(fact_8745_ATP_Olambda__565,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ane(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu)) ) ) ) ).

% ATP.lambda_565
tff(fact_8746_ATP_Olambda__566,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_and(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => ( ( Uub != X4 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_566
tff(fact_8747_ATP_Olambda__567,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ari(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => ( ( Uub != X4 )
              & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu)) ) ) ) ) ).

% ATP.lambda_567
tff(fact_8748_ATP_Olambda__568,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_to(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))),Uua)) ) ).

% ATP.lambda_568
tff(fact_8749_ATP_Olambda__569,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aiz(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).

% ATP.lambda_569
tff(fact_8750_ATP_Olambda__570,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_ajs(set(A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,Uu,Uua)),Uub)) ) ) ) ).

% ATP.lambda_570
tff(fact_8751_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_asx(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
            & ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_571
tff(fact_8752_ATP_Olambda__572,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_art(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( order_ofilter(A,Uu,Uua)
        & ( Uua != field2(A,Uu) )
        & order_ofilter(A,Uu,Uub)
        & ( Uub != field2(A,Uu) )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub)) ) ) ).

% ATP.lambda_572
tff(fact_8753_ATP_Olambda__573,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_alw(set(nat),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,Uub),Uu))
        & pp(aa(set(nat),bool,member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_alv(set(nat),fun(nat,fun(nat,bool)),Uu),Uub)))),Uua)) ) ) ).

% ATP.lambda_573
tff(fact_8754_ATP_Olambda__574,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(nat,fun(set(A),bool),aTP_Lamp_mf(set(A),fun(nat,fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu))
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_574
tff(fact_8755_ATP_Olambda__575,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_mn(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(nat,nat,suc,Uua))) ) ) ).

% ATP.lambda_575
tff(fact_8756_ATP_Olambda__576,axiom,
    ! [A: $tType,Uu: multiset(A),Uua: multiset(A),Uub: A] :
      ( pp(aa(A,bool,aa(multiset(A),fun(A,bool),aTP_Lamp_akj(multiset(A),fun(multiset(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uu),Uub))) ) ).

% ATP.lambda_576
tff(fact_8757_ATP_Olambda__577,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A)),Uub: fun(B,A)] : aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_ajg(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),Uu),Uua),Uub) = aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),inv_image(A,B,Uu,Uub)),inv_image(A,B,Uua,Uub)) ).

% ATP.lambda_577
tff(fact_8758_ATP_Olambda__578,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hb(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_578
tff(fact_8759_ATP_Olambda__579,axiom,
    ! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_cy(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uu),Uub))
        | pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uua),Uub)) ) ) ).

% ATP.lambda_579
tff(fact_8760_ATP_Olambda__580,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aj(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        | pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_580
tff(fact_8761_ATP_Olambda__581,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_eo(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_581
tff(fact_8762_ATP_Olambda__582,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ew(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_582
tff(fact_8763_ATP_Olambda__583,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_lt(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_583
tff(fact_8764_ATP_Olambda__584,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_eu(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_584
tff(fact_8765_ATP_Olambda__585,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_lv(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_585
tff(fact_8766_ATP_Olambda__586,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_lw(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_586
tff(fact_8767_ATP_Olambda__587,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_lu(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_587
tff(fact_8768_ATP_Olambda__588,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ajf(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_588
tff(fact_8769_ATP_Olambda__589,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_aje(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uua))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).

% ATP.lambda_589
tff(fact_8770_ATP_Olambda__590,axiom,
    ! [Uu: assn,Uua: assn,Uub: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aa(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool),aTP_Lamp_cz(assn,fun(assn,fun(product_prod(heap_ext(product_unit),set(nat)),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uu),Uub))
        & pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(Uua),Uub)) ) ) ).

% ATP.lambda_590
tff(fact_8771_ATP_Olambda__591,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_agx(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uu)) ) ) ).

% ATP.lambda_591
tff(fact_8772_ATP_Olambda__592,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_agu(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uu)) ) ) ).

% ATP.lambda_592
tff(fact_8773_ATP_Olambda__593,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_aqq(filter(A),fun(filter(A),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ( eventually(A,Uub,Uu)
        & eventually(A,Uub,Uua) ) ) ).

% ATP.lambda_593
tff(fact_8774_ATP_Olambda__594,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_anf(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uua),Uu))
        & pp(aa(set(A),bool,member(A,Uub),Uu)) ) ) ).

% ATP.lambda_594
tff(fact_8775_ATP_Olambda__595,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_alv(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,member(nat,Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_595
tff(fact_8776_ATP_Olambda__596,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A,Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),aTP_Lamp_atp(set(set(A)),fun(A,fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(A)),bool,member(set(A),Uub),Uu))
        & pp(aa(set(A),bool,member(A,Uua),Uub)) ) ) ).

% ATP.lambda_596
tff(fact_8777_ATP_Olambda__597,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_cq(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_597
tff(fact_8778_ATP_Olambda__598,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_auq(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uu )
        & ( Uub = Uu ) ) ) ).

% ATP.lambda_598
tff(fact_8779_ATP_Olambda__599,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ki(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uub),Uu) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_599
tff(fact_8780_ATP_Olambda__600,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_tx(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_600
tff(fact_8781_ATP_Olambda__601,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_tv(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_601
tff(fact_8782_ATP_Olambda__602,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_apq(fun(A,bool),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uua))
       => pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_602
tff(fact_8783_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: multiset(A),Uub: A] : aa(A,fun(B,B),aa(multiset(A),fun(A,fun(B,B)),aTP_Lamp_alb(fun(A,fun(B,B)),fun(multiset(A),fun(A,fun(B,B))),Uu),Uua),Uub) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_603
tff(fact_8784_ATP_Olambda__604,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_kv(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_604
tff(fact_8785_ATP_Olambda__605,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ajw(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_605
tff(fact_8786_ATP_Olambda__606,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_af(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_606
tff(fact_8787_ATP_Olambda__607,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_uv(fun(A,bool),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uua))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_607
tff(fact_8788_ATP_Olambda__608,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ux(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_608
tff(fact_8789_ATP_Olambda__609,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_er(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_609
tff(fact_8790_ATP_Olambda__610,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_eq(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_610
tff(fact_8791_ATP_Olambda__611,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_abv(set(A),fun(fun(A,set(B)),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_611
tff(fact_8792_ATP_Olambda__612,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_kr(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
            & ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_612
tff(fact_8793_ATP_Olambda__613,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_613
tff(fact_8794_ATP_Olambda__614,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_kt(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
            & ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_614
tff(fact_8795_ATP_Olambda__615,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_mg(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uua))
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_615
tff(fact_8796_ATP_Olambda__616,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_pq(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uua))
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_616
tff(fact_8797_ATP_Olambda__617,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_mb(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,member(B,Uub),Uu))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one))),aa(B,A,Uua,Uub))) ) ) ) ).

% ATP.lambda_617
tff(fact_8798_ATP_Olambda__618,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_dc(array(A),fun(list(A),fun(nat,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uu,Uua)),pure_assn(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))) ) ).

% ATP.lambda_618
tff(fact_8799_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_cd(ref(A),fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,Uu,Uua)),pure_assn(aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua))) ) ).

% ATP.lambda_619
tff(fact_8800_ATP_Olambda__620,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_cn(array(A),fun(list(A),fun(list(A),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),snga_assn(A,Uu,Uua)),pure_assn(aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Uub),Uua))) ) ).

% ATP.lambda_620
tff(fact_8801_ATP_Olambda__621,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_di(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
        & ~ pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_621
tff(fact_8802_ATP_Olambda__622,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: multiset(A),Uub: A] : aa(A,nat,aa(multiset(A),fun(A,nat),aTP_Lamp_akq(fun(A,nat),fun(multiset(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Uua),Uub)),aa(nat,nat,suc,aa(A,nat,Uu,Uub))) ).

% ATP.lambda_622
tff(fact_8803_ATP_Olambda__623,axiom,
    ! [A: $tType,D: $tType,C: $tType,Uu: fun(C,fun(D,bool)),Uua: fun(C,A),Uub: set(A)] : aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),aTP_Lamp_apm(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uua),Uub)),aa(fun(C,bool),set(C),collect(C),aa(fun(C,fun(D,bool)),fun(C,bool),domainp(C,D),Uu))) ).

% ATP.lambda_623
tff(fact_8804_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_uh(list(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_624
tff(fact_8805_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_agw(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uu)),Uua)) ) ).

% ATP.lambda_625
tff(fact_8806_ATP_Olambda__626,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_gy(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_626
tff(fact_8807_ATP_Olambda__627,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jl(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_627
tff(fact_8808_ATP_Olambda__628,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wa(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_628
tff(fact_8809_ATP_Olambda__629,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vy(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_629
tff(fact_8810_ATP_Olambda__630,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vz(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_630
tff(fact_8811_ATP_Olambda__631,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vx(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_631
tff(fact_8812_ATP_Olambda__632,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
      ( pp(aa(C,bool,aa(B,fun(C,bool),aTP_Lamp_ark(set(product_prod(B,C)),fun(B,fun(C,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)),Uu)) ) ).

% ATP.lambda_632
tff(fact_8813_ATP_Olambda__633,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_at(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_633
tff(fact_8814_ATP_Olambda__634,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_ao(set(product_prod(A,B)),fun(A,fun(B,bool))),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,B)),bool,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_634
tff(fact_8815_ATP_Olambda__635,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bc(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_635
tff(fact_8816_ATP_Olambda__636,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_akb(set(product_prod(B,A)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua)),Uu)) ) ).

% ATP.lambda_636
tff(fact_8817_ATP_Olambda__637,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ake(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ).

% ATP.lambda_637
tff(fact_8818_ATP_Olambda__638,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: A,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_qt(set(list(A)),fun(A,fun(list(A),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_638
tff(fact_8819_ATP_Olambda__639,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_tf(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_639
tff(fact_8820_ATP_Olambda__640,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_la(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,member(A,Uub),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uub))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uub),Uua)) ) ) ) ).

% ATP.lambda_640
tff(fact_8821_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_qh(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))))),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_641
tff(fact_8822_ATP_Olambda__642,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,A),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,A),fun(A,bool),aTP_Lamp_ata(fun(A,B),fun(fun(B,A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uua,aa(A,B,Uu,Uub)) = Uub ) ) ).

% ATP.lambda_642
tff(fact_8823_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(B,bool),fun(product_prod(A,B),bool),aTP_Lamp_ath(fun(A,bool),fun(fun(B,bool),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uub)))
        & pp(aa(B,bool,Uua,aa(product_prod(A,B),B,product_snd(A,B),Uub))) ) ) ).

% ATP.lambda_643
tff(fact_8824_ATP_Olambda__644,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_kj(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,Uu,Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_644
tff(fact_8825_ATP_Olambda__645,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aph(fun(A,fun(B,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu),Uub)) ) ) ).

% ATP.lambda_645
tff(fact_8826_ATP_Olambda__646,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_qf(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))))) ) ) ).

% ATP.lambda_646
tff(fact_8827_ATP_Olambda__647,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aTP_Lamp_qg(fun(B,A),fun(list(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))) ) ) ) ).

% ATP.lambda_647
tff(fact_8828_ATP_Olambda__648,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_acc(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uua,Uub),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_648
tff(fact_8829_ATP_Olambda__649,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_acb(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uua,Uub),aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_649
tff(fact_8830_ATP_Olambda__650,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_rz(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_650
tff(fact_8831_ATP_Olambda__651,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqf(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_651
tff(fact_8832_ATP_Olambda__652,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_ik(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_652
tff(fact_8833_ATP_Olambda__653,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,B)),Uua: fun(A,option(C)),Uub: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aTP_Lamp_aic(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),Uu),Uua),Uub) = aa(option(C),option(B),map_option(C,B,aa(A,fun(C,B),Uu,Uub)),aa(A,option(C),Uua,Uub)) ).

% ATP.lambda_653
tff(fact_8834_ATP_Olambda__654,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_ij(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_654
tff(fact_8835_ATP_Olambda__655,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_ft(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_655
tff(fact_8836_ATP_Olambda__656,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_ge(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_656
tff(fact_8837_ATP_Olambda__657,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_abk(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_657
tff(fact_8838_ATP_Olambda__658,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_aky(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_658
tff(fact_8839_ATP_Olambda__659,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_jr(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_659
tff(fact_8840_ATP_Olambda__660,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_gh(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_660
tff(fact_8841_ATP_Olambda__661,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_xr(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_661
tff(fact_8842_ATP_Olambda__662,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_arq(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_662
tff(fact_8843_ATP_Olambda__663,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_xg(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_663
tff(fact_8844_ATP_Olambda__664,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_arp(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_664
tff(fact_8845_ATP_Olambda__665,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_bn(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_665
tff(fact_8846_ATP_Olambda__666,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_abp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_666
tff(fact_8847_ATP_Olambda__667,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_yx(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_667
tff(fact_8848_ATP_Olambda__668,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ast(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_668
tff(fact_8849_ATP_Olambda__669,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_yv(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_669
tff(fact_8850_ATP_Olambda__670,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ass(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_670
tff(fact_8851_ATP_Olambda__671,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_abn(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_671
tff(fact_8852_ATP_Olambda__672,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_fz(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_672
tff(fact_8853_ATP_Olambda__673,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_akx(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_673
tff(fact_8854_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_any(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),aa(B,multiset(A),Uu,Uub)),aa(B,multiset(A),Uua,Uub)) ).

% ATP.lambda_674
tff(fact_8855_ATP_Olambda__675,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,multiset(A)),Uua: fun(B,multiset(A)),Uub: B] : aa(B,multiset(A),aa(fun(B,multiset(A)),fun(B,multiset(A)),aTP_Lamp_ald(fun(B,multiset(A)),fun(fun(B,multiset(A)),fun(B,multiset(A))),Uu),Uua),Uub) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),aa(B,multiset(A),Uu,Uub)),aa(B,multiset(A),Uua,Uub)) ).

% ATP.lambda_675
tff(fact_8856_ATP_Olambda__676,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_afc(fun(C,A),fun(fun(C,B),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),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_676
tff(fact_8857_ATP_Olambda__677,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ate(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).

% ATP.lambda_677
tff(fact_8858_ATP_Olambda__678,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_dd(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_678
tff(fact_8859_ATP_Olambda__679,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aom(fun(A,set(B)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),aa(A,set(B),Uu,Uua)),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_679
tff(fact_8860_ATP_Olambda__680,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: fun(A,nat),Uub: A] : aa(A,fun(B,B),aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_uy(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),Uu),Uua),Uub) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(A,nat,Uua,Uub)),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_680
tff(fact_8861_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ara(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(A,bool,Uu,Uub))
            | pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_681
tff(fact_8862_ATP_Olambda__682,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ai(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_682
tff(fact_8863_ATP_Olambda__683,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_op(fun(B,bool),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(B,bool,Uu,Uub))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_683
tff(fact_8864_ATP_Olambda__684,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ah(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_684
tff(fact_8865_ATP_Olambda__685,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qi(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_685
tff(fact_8866_ATP_Olambda__686,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_sa(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uua) = aa(B,A,Uu,Uub) ) ) ) ).

% ATP.lambda_686
tff(fact_8867_ATP_Olambda__687,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ahp(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_687
tff(fact_8868_ATP_Olambda__688,axiom,
    ! [D: $tType,B: $tType,Uu: fun(B,D),Uua: fun(B,D),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,D),fun(B,bool),aTP_Lamp_avd(fun(B,D),fun(fun(B,D),fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,D,Uu,Uub) = aa(B,D,Uua,Uub) ) ) ).

% ATP.lambda_688
tff(fact_8869_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqd(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_689
tff(fact_8870_ATP_Olambda__690,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aqa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
      <=> pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_690
tff(fact_8871_ATP_Olambda__691,axiom,
    ! [C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(A,C),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,C),fun(A,bool),aTP_Lamp_avc(fun(A,C),fun(fun(A,C),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,C,Uu,Uub) = aa(A,C,Uua,Uub) ) ) ).

% ATP.lambda_691
tff(fact_8872_ATP_Olambda__692,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ape(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_692
tff(fact_8873_ATP_Olambda__693,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ub(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_693
tff(fact_8874_ATP_Olambda__694,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_kg(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uua,Uub))) ) ).

% ATP.lambda_694
tff(fact_8875_ATP_Olambda__695,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ali(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & ! [Y4: A] :
            ( pp(aa(A,bool,Uua,Y4))
           => pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y4)) ) ) ) ).

% ATP.lambda_695
tff(fact_8876_ATP_Olambda__696,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_bm(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_696
tff(fact_8877_ATP_Olambda__697,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_eg(fun(A,option(B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(A,option(B),Uu,Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_697
tff(fact_8878_ATP_Olambda__698,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_ig(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_698
tff(fact_8879_ATP_Olambda__699,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_fw(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_699
tff(fact_8880_ATP_Olambda__700,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_kk(fun(nat,nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua)) ) ).

% ATP.lambda_700
tff(fact_8881_ATP_Olambda__701,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_xk(fun(B,set(A)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_701
tff(fact_8882_ATP_Olambda__702,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aql(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_702
tff(fact_8883_ATP_Olambda__703,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aqi(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_703
tff(fact_8884_ATP_Olambda__704,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_fo(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_704
tff(fact_8885_ATP_Olambda__705,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_lm(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_705
tff(fact_8886_ATP_Olambda__706,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_gg(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_706
tff(fact_8887_ATP_Olambda__707,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_qu(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_707
tff(fact_8888_ATP_Olambda__708,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aqj(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_708
tff(fact_8889_ATP_Olambda__709,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_gc(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_709
tff(fact_8890_ATP_Olambda__710,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_cj(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_710
tff(fact_8891_ATP_Olambda__711,axiom,
    ! [K6: $tType,L8: $tType,Uu: fun(K6,set(L8)),Uua: set(L8),Uub: K6] : aa(K6,set(L8),aa(set(L8),fun(K6,set(L8)),aTP_Lamp_xs(fun(K6,set(L8)),fun(set(L8),fun(K6,set(L8))),Uu),Uua),Uub) = aa(set(L8),set(L8),aa(set(L8),fun(set(L8),set(L8)),minus_minus(set(L8)),aa(K6,set(L8),Uu,Uub)),Uua) ).

% ATP.lambda_711
tff(fact_8892_ATP_Olambda__712,axiom,
    ! [E: $tType,F2: $tType,Uu: fun(E,set(F2)),Uua: set(F2),Uub: E] : aa(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_zw(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),Uu),Uua),Uub) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(E,set(F2),Uu,Uub)),Uua) ).

% ATP.lambda_712
tff(fact_8893_ATP_Olambda__713,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_il(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_713
tff(fact_8894_ATP_Olambda__714,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,multiset(A)),Uua: A,Uub: B] : aa(B,nat,aa(A,fun(B,nat),aTP_Lamp_akh(fun(B,multiset(A)),fun(A,fun(B,nat)),Uu),Uua),Uub) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(B,multiset(A),Uu,Uub)),Uua) ).

% ATP.lambda_714
tff(fact_8895_ATP_Olambda__715,axiom,
    ! [K6: $tType,L8: $tType,Uu: fun(K6,set(L8)),Uua: set(L8),Uub: K6] : aa(K6,set(L8),aa(set(L8),fun(K6,set(L8)),aTP_Lamp_yz(fun(K6,set(L8)),fun(set(L8),fun(K6,set(L8))),Uu),Uua),Uub) = aa(set(L8),set(L8),aa(set(L8),fun(set(L8),set(L8)),sup_sup(set(L8)),aa(K6,set(L8),Uu,Uub)),Uua) ).

% ATP.lambda_715
tff(fact_8896_ATP_Olambda__716,axiom,
    ! [C: $tType,D: $tType,Uu: fun(C,set(D)),Uua: set(D),Uub: C] : aa(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_ww(fun(C,set(D)),fun(set(D),fun(C,set(D))),Uu),Uua),Uub) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(C,set(D),Uu,Uub)),Uua) ).

% ATP.lambda_716
tff(fact_8897_ATP_Olambda__717,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_yt(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_717
tff(fact_8898_ATP_Olambda__718,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_bo(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_718
tff(fact_8899_ATP_Olambda__719,axiom,
    ! [G: $tType,H10: $tType,Uu: fun(G,set(H10)),Uua: set(H10),Uub: G] : aa(G,set(H10),aa(set(H10),fun(G,set(H10)),aTP_Lamp_xp(fun(G,set(H10)),fun(set(H10),fun(G,set(H10))),Uu),Uua),Uub) = aa(set(H10),set(H10),aa(set(H10),fun(set(H10),set(H10)),inf_inf(set(H10)),aa(G,set(H10),Uu,Uub)),Uua) ).

% ATP.lambda_719
tff(fact_8900_ATP_Olambda__720,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_zn(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_720
tff(fact_8901_ATP_Olambda__721,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ze(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_721
tff(fact_8902_ATP_Olambda__722,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_ct(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_722
tff(fact_8903_ATP_Olambda__723,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_zi(fun(A,set(B)),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),Uua) ).

% ATP.lambda_723
tff(fact_8904_ATP_Olambda__724,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_agv(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_724
tff(fact_8905_ATP_Olambda__725,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,heap_Time_Heap(C)),Uua: fun(C,heap_Time_Heap(B)),Uub: A] : aa(A,heap_Time_Heap(B),aa(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aTP_Lamp_ae(fun(A,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),Uu),Uua),Uub) = heap_Time_bind(C,B,aa(A,heap_Time_Heap(C),Uu,Uub),Uua) ).

% ATP.lambda_725
tff(fact_8906_ATP_Olambda__726,axiom,
    ! [E: $tType,F2: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F2,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F2,B),aa(fun(C,fun(D,B)),fun(E,fun(F2,B)),aTP_Lamp_fb(fun(E,fun(F2,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F2,B))),Uu),Uua),Uub) = product_scomp(F2,C,D,B,aa(E,fun(F2,product_prod(C,D)),Uu,Uub),Uua) ).

% ATP.lambda_726
tff(fact_8907_ATP_Olambda__727,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_atm(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).

% ATP.lambda_727
tff(fact_8908_ATP_Olambda__728,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,fun(A,fun(C,bool))),Uua: fun(C,fun(B,bool)),Uub: D] : aa(D,fun(A,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool))),aTP_Lamp_arm(fun(D,fun(A,fun(C,bool))),fun(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool)))),Uu),Uua),Uub) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(D,fun(A,fun(C,bool)),Uu,Uub)),Uua) ).

% ATP.lambda_728
tff(fact_8909_ATP_Olambda__729,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(A,C))),Uua: set(product_prod(C,B)),Uub: D] : aa(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_xe(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,aa(D,set(product_prod(A,C)),Uu,Uub),Uua) ).

% ATP.lambda_729
tff(fact_8910_ATP_Olambda__730,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: set(B),Uub: C] : aa(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_aiq(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,aa(C,set(product_prod(B,A)),Uu,Uub)),Uua) ).

% ATP.lambda_730
tff(fact_8911_ATP_Olambda__731,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_afg(fun(A,B),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(B),bool,member(B,aa(A,B,Uu,Uub)),Uua)) ) ).

% ATP.lambda_731
tff(fact_8912_ATP_Olambda__732,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ahi(set(A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,aa(B,A,Uua,Uub)),Uu)) ) ).

% ATP.lambda_732
tff(fact_8913_ATP_Olambda__733,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: bool,Uub: A] :
      ( pp(aa(A,bool,aa(bool,fun(A,bool),aTP_Lamp_apz(fun(A,bool),fun(bool,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(Uua) ) ) ).

% ATP.lambda_733
tff(fact_8914_ATP_Olambda__734,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_qv(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_734
tff(fact_8915_ATP_Olambda__735,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_qk(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_735
tff(fact_8916_ATP_Olambda__736,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ahv(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_736
tff(fact_8917_ATP_Olambda__737,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ti(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rj(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,aa(list(nat),list(A),map(nat,A,Uua),upt(zero_zero(nat),Uu)),Uub)) ) ).

% ATP.lambda_737
tff(fact_8918_ATP_Olambda__738,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rm(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rj(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,replicate(A,Uu,Uua),Uub)) ) ).

% ATP.lambda_738
tff(fact_8919_ATP_Olambda__739,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aru(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uua )
        & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ) ).

% ATP.lambda_739
tff(fact_8920_ATP_Olambda__740,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ep(A,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_740
tff(fact_8921_ATP_Olambda__741,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iv(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_741
tff(fact_8922_ATP_Olambda__742,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_amg(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua != Uu )
        & ( Uub != Uu ) ) ) ).

% ATP.lambda_742
tff(fact_8923_ATP_Olambda__743,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kf(fun(B,bool),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uub))),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_743
tff(fact_8924_ATP_Olambda__744,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ang(fun(A,A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( ? [X4: A] :
                ( ( Uub = aa(A,A,Uu,X4) )
                & pp(aa(A,bool,Uua,X4)) )
            | ? [M13: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M13) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M13)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),M13))
                   => pp(aa(A,bool,Uua,X4)) ) ) ) ) ) ).

% ATP.lambda_744
tff(fact_8925_ATP_Olambda__745,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(set(B),fun(fun(A,B),bool),aTP_Lamp_amq(set(A),fun(set(B),fun(fun(A,B),bool)),Uu),Uua),Uub))
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uu))
           => pp(aa(set(B),bool,member(B,aa(A,B,Uub,X4)),Uua)) )
        & ! [A9: A] :
            ( ~ pp(aa(set(A),bool,member(A,A9),Uu))
           => ( aa(A,B,Uub,A9) = undefined(B) ) ) ) ) ).

% ATP.lambda_745
tff(fact_8926_ATP_Olambda__746,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),bool)),Uua: list(A),Uub: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_afx(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4)) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X4))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs3),Ys4)) ) ) ) ) ).

% ATP.lambda_746
tff(fact_8927_ATP_Olambda__747,axiom,
    ! [A: $tType,Uu: fun(list(multiset(A)),fun(list(multiset(A)),bool)),Uua: list(multiset(A)),Uub: list(multiset(A))] :
      ( pp(aa(list(multiset(A)),bool,aa(list(multiset(A)),fun(list(multiset(A)),bool),aa(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool)),aTP_Lamp_atu(fun(list(multiset(A)),fun(list(multiset(A)),bool)),fun(list(multiset(A)),fun(list(multiset(A)),bool))),Uu),Uua),Uub))
    <=> ( ? [Y4: multiset(A),Ys4: list(multiset(A))] :
            ( ( Uua = nil(multiset(A)) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y4),Ys4) ) )
        | ? [X4: multiset(A),Y4: multiset(A),Xs3: list(multiset(A)),Ys4: list(multiset(A))] :
            ( ( Uua = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),X4),Xs3) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y4),Ys4) )
            & pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),X4),Y4)) )
        | ? [X4: multiset(A),Y4: multiset(A),Xs3: list(multiset(A)),Ys4: list(multiset(A))] :
            ( ( Uua = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),X4),Xs3) )
            & ( Uub = aa(list(multiset(A)),list(multiset(A)),aa(multiset(A),fun(list(multiset(A)),list(multiset(A))),cons(multiset(A)),Y4),Ys4) )
            & ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),X4),Y4))
            & ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subset_mset(A),Y4),X4))
            & pp(aa(list(multiset(A)),bool,aa(list(multiset(A)),fun(list(multiset(A)),bool),Uu,Xs3),Ys4)) ) ) ) ).

% ATP.lambda_747
tff(fact_8928_ATP_Olambda__748,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_aoh(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_748
tff(fact_8929_ATP_Olambda__749,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_alk(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
        & pp(aa(set(A),bool,finite_finite2(A),Uub))
        & ( Uub != bot_bot(set(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),Uua))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),Uub))
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),Uu)) ) ) ) ) ).

% ATP.lambda_749
tff(fact_8930_ATP_Olambda__750,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_akd(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),image(A,A,converse(A,A,Uu)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A)))) ).

% ATP.lambda_750
tff(fact_8931_ATP_Olambda__751,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B] : aa(B,fun(list(B),list(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_py(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua),Uub) = case_list(list(B),B,Uua,aa(B,fun(B,fun(list(B),list(B))),aa(list(B),fun(B,fun(B,fun(list(B),list(B)))),aTP_Lamp_px(fun(B,A),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub)) ) ).

% ATP.lambda_751
tff(fact_8932_ATP_Olambda__752,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_jt(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_752
tff(fact_8933_ATP_Olambda__753,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_js(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = 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_753
tff(fact_8934_ATP_Olambda__754,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: list(B),Uub: A] : aa(A,list(B),aa(list(B),fun(A,list(B)),aTP_Lamp_se(fun(A,B),fun(list(B),fun(A,list(B))),Uu),Uua),Uub) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uua),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,Uu,Uub)),nil(B))) ).

% ATP.lambda_754
tff(fact_8935_ATP_Olambda__755,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hx(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_755
tff(fact_8936_ATP_Olambda__756,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ht(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_756
tff(fact_8937_ATP_Olambda__757,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rj(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),one_one(nat)))) ) ).

% ATP.lambda_757
tff(fact_8938_ATP_Olambda__758,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_anl(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A)))))) ) ).

% ATP.lambda_758
tff(fact_8939_ATP_Olambda__759,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ka(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Uu)))) ) ).

% ATP.lambda_759
tff(fact_8940_ATP_Olambda__760,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_te(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,aa(list(nat),list(A),map(nat,A,Uua),upt(zero_zero(nat),Uu))) ) ).

% ATP.lambda_760
tff(fact_8941_ATP_Olambda__761,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_jy(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_761
tff(fact_8942_ATP_Olambda__762,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aih(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),dom(A,B,Uu)))) ) ).

% ATP.lambda_762
tff(fact_8943_ATP_Olambda__763,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aii(fun(A,option(B)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu)))) ) ).

% ATP.lambda_763
tff(fact_8944_ATP_Olambda__764,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] :
          ( pp(aa(heap_ext(product_unit),bool,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_aos(nat,fun(array(A),fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),array_length(A,Uub,Uua))) ) ) ).

% ATP.lambda_764
tff(fact_8945_ATP_Olambda__765,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] :
          ( pp(aa(heap_ext(product_unit),bool,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_aoo(array(A),fun(nat,fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),array_length(A,Uub,Uu))) ) ) ).

% ATP.lambda_765
tff(fact_8946_ATP_Olambda__766,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_fr(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_766
tff(fact_8947_ATP_Olambda__767,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hv(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_767
tff(fact_8948_ATP_Olambda__768,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_rk(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,replicate(A,Uu,Uua)) ) ).

% ATP.lambda_768
tff(fact_8949_ATP_Olambda__769,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_sn(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_769
tff(fact_8950_ATP_Olambda__770,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_sr(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_770
tff(fact_8951_ATP_Olambda__771,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_aas(A,fun(list(A),fun(nat,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),take(A,Uub,Uua)) ).

% ATP.lambda_771
tff(fact_8952_ATP_Olambda__772,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_sy(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),map_filter(B,A,Uu,Uua)) ).

% ATP.lambda_772
tff(fact_8953_ATP_Olambda__773,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ajx(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(B),bool,member(B,Uub),aa(set(A),set(B),image2(A,B,Uu),Uua))) ) ) ).

% ATP.lambda_773
tff(fact_8954_ATP_Olambda__774,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ya(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(C,A,Uua,aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_774
tff(fact_8955_ATP_Olambda__775,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_xz(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = groups7311177749621191930dd_sum(C,A,Uua,aa(B,set(C),Uu,Uub)) ) ).

% ATP.lambda_775
tff(fact_8956_ATP_Olambda__776,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aqk(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_776
tff(fact_8957_ATP_Olambda__777,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aqg(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_777
tff(fact_8958_ATP_Olambda__778,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_qw(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_778
tff(fact_8959_ATP_Olambda__779,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aqh(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_779
tff(fact_8960_ATP_Olambda__780,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,option(C)),Uub: B] : aa(B,option(A),aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_vs(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),Uu),Uua),Uub) = aa(option(C),option(A),map_option(C,A,Uu),aa(B,option(C),Uua,Uub)) ).

% ATP.lambda_780
tff(fact_8961_ATP_Olambda__781,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(nat,fun(fun(A,nat),fun(A,nat)),aTP_Lamp_ala(nat,fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_781
tff(fact_8962_ATP_Olambda__782,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_gd(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_782
tff(fact_8963_ATP_Olambda__783,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: nat,Uub: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_anh(fun(A,nat),fun(nat,fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_783
tff(fact_8964_ATP_Olambda__784,axiom,
    ! [M10: $tType,N9: $tType,Uu: set(M10),Uua: fun(N9,set(M10)),Uub: N9] : aa(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_zj(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),Uu),Uua),Uub) = aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),minus_minus(set(M10)),Uu),aa(N9,set(M10),Uua,Uub)) ).

% ATP.lambda_784
tff(fact_8965_ATP_Olambda__785,axiom,
    ! [G: $tType,H10: $tType,Uu: set(G),Uua: fun(H10,set(G)),Uub: H10] : aa(H10,set(G),aa(fun(H10,set(G)),fun(H10,set(G)),aTP_Lamp_zm(set(G),fun(fun(H10,set(G)),fun(H10,set(G))),Uu),Uua),Uub) = aa(set(G),set(G),aa(set(G),fun(set(G),set(G)),minus_minus(set(G)),Uu),aa(H10,set(G),Uua,Uub)) ).

% ATP.lambda_785
tff(fact_8966_ATP_Olambda__786,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_it(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_786
tff(fact_8967_ATP_Olambda__787,axiom,
    ! [M10: $tType,N9: $tType,Uu: set(M10),Uua: fun(N9,set(M10)),Uub: N9] : aa(N9,set(M10),aa(fun(N9,set(M10)),fun(N9,set(M10)),aTP_Lamp_yy(set(M10),fun(fun(N9,set(M10)),fun(N9,set(M10))),Uu),Uua),Uub) = aa(set(M10),set(M10),aa(set(M10),fun(set(M10),set(M10)),sup_sup(set(M10)),Uu),aa(N9,set(M10),Uua,Uub)) ).

% ATP.lambda_787
tff(fact_8968_ATP_Olambda__788,axiom,
    ! [E: $tType,F2: $tType,Uu: set(E),Uua: fun(F2,set(E)),Uub: F2] : aa(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_wv(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),Uu),Uua),Uub) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),Uu),aa(F2,set(E),Uua,Uub)) ).

% ATP.lambda_788
tff(fact_8969_ATP_Olambda__789,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_za(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_789
tff(fact_8970_ATP_Olambda__790,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_yr(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_790
tff(fact_8971_ATP_Olambda__791,axiom,
    ! [I6: $tType,J5: $tType,Uu: set(I6),Uua: fun(J5,set(I6)),Uub: J5] : aa(J5,set(I6),aa(fun(J5,set(I6)),fun(J5,set(I6)),aTP_Lamp_xq(set(I6),fun(fun(J5,set(I6)),fun(J5,set(I6))),Uu),Uua),Uub) = aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),inf_inf(set(I6)),Uu),aa(J5,set(I6),Uua,Uub)) ).

% ATP.lambda_791
tff(fact_8972_ATP_Olambda__792,axiom,
    ! [C: $tType,D: $tType,Uu: set(C),Uua: fun(D,set(C)),Uub: D] : aa(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_zh(set(C),fun(fun(D,set(C)),fun(D,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),Uu),aa(D,set(C),Uua,Uub)) ).

% ATP.lambda_792
tff(fact_8973_ATP_Olambda__793,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_xo(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_793
tff(fact_8974_ATP_Olambda__794,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_zp(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_794
tff(fact_8975_ATP_Olambda__795,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zf(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_795
tff(fact_8976_ATP_Olambda__796,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ln(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_796
tff(fact_8977_ATP_Olambda__797,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_atn(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).

% ATP.lambda_797
tff(fact_8978_ATP_Olambda__798,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,filter(C)),Uub: B] : aa(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_arb(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,C,Uu,aa(B,filter(C),Uua,Uub)) ).

% ATP.lambda_798
tff(fact_8979_ATP_Olambda__799,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aqz(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_799
tff(fact_8980_ATP_Olambda__800,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(D,fun(C,fun(B,bool))),Uub: D] : aa(D,fun(A,fun(B,bool)),aa(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool))),aTP_Lamp_arl(fun(A,fun(C,bool)),fun(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool)))),Uu),Uua),Uub) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),Uu),aa(D,fun(C,fun(B,bool)),Uua,Uub)) ).

% ATP.lambda_800
tff(fact_8981_ATP_Olambda__801,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_rw(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_801
tff(fact_8982_ATP_Olambda__802,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: set(product_prod(A,C)),Uua: fun(D,set(product_prod(C,B))),Uub: D] : aa(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_xd(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,Uu,aa(D,set(product_prod(C,B)),Uua,Uub)) ).

% ATP.lambda_802
tff(fact_8983_ATP_Olambda__803,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_asz(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtermap(B,A,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_803
tff(fact_8984_ATP_Olambda__804,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_aio(set(product_prod(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_804
tff(fact_8985_ATP_Olambda__805,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_apx(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_805
tff(fact_8986_ATP_Olambda__806,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_afh(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_806
tff(fact_8987_ATP_Olambda__807,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_yu(fun(B,set(A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,Uua))) ) ).

% ATP.lambda_807
tff(fact_8988_ATP_Olambda__808,axiom,
    ! [I6: $tType,J5: $tType,Uu: I6,Uua: fun(J5,set(I6)),Uub: J5] : aa(J5,set(I6),aa(fun(J5,set(I6)),fun(J5,set(I6)),aTP_Lamp_yw(I6,fun(fun(J5,set(I6)),fun(J5,set(I6))),Uu),Uua),Uub) = aa(set(I6),set(I6),aa(I6,fun(set(I6),set(I6)),insert(I6),Uu),aa(J5,set(I6),Uua,Uub)) ).

% ATP.lambda_808
tff(fact_8989_ATP_Olambda__809,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_xl(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uu),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_809
tff(fact_8990_ATP_Olambda__810,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_wu(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_810
tff(fact_8991_ATP_Olambda__811,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_xh(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_811
tff(fact_8992_ATP_Olambda__812,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_ahr(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image2(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_812
tff(fact_8993_ATP_Olambda__813,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_arn(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub)) ) ).

% ATP.lambda_813
tff(fact_8994_ATP_Olambda__814,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_apy(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_814
tff(fact_8995_ATP_Olambda__815,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_alo(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        | pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_815
tff(fact_8996_ATP_Olambda__816,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_amn(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        & pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_816
tff(fact_8997_ATP_Olambda__817,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_qn(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_817
tff(fact_8998_ATP_Olambda__818,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ahn(fun(A,B),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_818
tff(fact_8999_ATP_Olambda__819,axiom,
    ! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_cm(assn,fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),pure_assn(aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua))) ).

% ATP.lambda_819
tff(fact_9000_ATP_Olambda__820,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jd(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_820
tff(fact_9001_ATP_Olambda__821,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_tp(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(list(A),set(A),set2(A),nths(A,Uu,Uua)))) ) ).

% ATP.lambda_821
tff(fact_9002_ATP_Olambda__822,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: A] : aa(A,set(A),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_arz(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uua),field2(B,Uu)) ).

% ATP.lambda_822
tff(fact_9003_ATP_Olambda__823,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(A),Uua: set(C),Uub: fun(A,B)] : aa(fun(A,B),set(fun(A,C)),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_asb(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),Uu),Uua),Uub) = bNF_Wellorder_Func(A,C,Uu,Uua) ).

% ATP.lambda_823
tff(fact_9004_ATP_Olambda__824,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] : aa(A,set(A),aa(set(A),fun(A,set(A)),aTP_Lamp_afa(set(A),fun(set(A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),Uua) ).

% ATP.lambda_824
tff(fact_9005_ATP_Olambda__825,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_abm(set(B),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu),Uua) ).

% ATP.lambda_825
tff(fact_9006_ATP_Olambda__826,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_arw(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).

% ATP.lambda_826
tff(fact_9007_ATP_Olambda__827,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_ajc(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_above(A,Uu,Uua) ).

% ATP.lambda_827
tff(fact_9008_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: A,Uub: product_unit] : aa(product_unit,assn,aa(A,fun(product_unit,assn),aTP_Lamp_by(ref(A),fun(A,fun(product_unit,assn)),Uu),Uua),Uub) = sngr_assn(A,Uu,Uua) ) ).

% ATP.lambda_828
tff(fact_9009_ATP_Olambda__829,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: A] : aa(A,set(sum_sum(B,C)),aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_asm(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Uu),Uua),Uub) = sum_Plus(B,C,Uu,Uua) ).

% ATP.lambda_829
tff(fact_9010_ATP_Olambda__830,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_tl(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),Uu) ).

% ATP.lambda_830
tff(fact_9011_ATP_Olambda__831,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_abe(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uu),Uua) ).

% ATP.lambda_831
tff(fact_9012_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_abu(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uu),Uua) ).

% ATP.lambda_832
tff(fact_9013_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] : aa(A,set(A),aa(set(B),fun(A,set(A)),aTP_Lamp_acd(fun(B,A),fun(set(B),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),Uua) ).

% ATP.lambda_833
tff(fact_9014_ATP_Olambda__834,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lb(fun(A,nat),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),Uub),Uua),aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub)))) ) ).

% ATP.lambda_834
tff(fact_9015_ATP_Olambda__835,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ky(fun(A,nat),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),if(nat,aa(A,bool,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat)))) ) ).

% ATP.lambda_835
tff(fact_9016_ATP_Olambda__836,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: ref(A),Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_cw(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,Uu,Uua,Uub)),one_one(nat))) ) ).

% ATP.lambda_836
tff(fact_9017_ATP_Olambda__837,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,nat),fun(A,bool),aTP_Lamp_le(fun(A,nat),fun(fun(A,nat),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub)))) ) ).

% ATP.lambda_837
tff(fact_9018_ATP_Olambda__838,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_jo(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_838
tff(fact_9019_ATP_Olambda__839,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_jn(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_839
tff(fact_9020_ATP_Olambda__840,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_app(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_840
tff(fact_9021_ATP_Olambda__841,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_xv(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_841
tff(fact_9022_ATP_Olambda__842,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_ie(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_842
tff(fact_9023_ATP_Olambda__843,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_go(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_843
tff(fact_9024_ATP_Olambda__844,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_asy(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub))) ) ).

% ATP.lambda_844
tff(fact_9025_ATP_Olambda__845,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_dr(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_845
tff(fact_9026_ATP_Olambda__846,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_tb(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ).

% ATP.lambda_846
tff(fact_9027_ATP_Olambda__847,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(multiset(A),B),Uua: fun(C,multiset(A)),Uub: C] : aa(C,B,aa(fun(C,multiset(A)),fun(C,B),aTP_Lamp_akt(fun(multiset(A),B),fun(fun(C,multiset(A)),fun(C,B)),Uu),Uua),Uub) = aa(multiset(A),B,Uu,aa(C,multiset(A),Uua,Uub)) ) ).

% ATP.lambda_847
tff(fact_9028_ATP_Olambda__848,axiom,
    ! [V4: $tType,U5: $tType,T: $tType,Uu: fun(U5,set(V4)),Uua: fun(T,U5),Uub: T] : aa(T,set(V4),aa(fun(T,U5),fun(T,set(V4)),aTP_Lamp_xi(fun(U5,set(V4)),fun(fun(T,U5),fun(T,set(V4))),Uu),Uua),Uub) = aa(U5,set(V4),Uu,aa(T,U5,Uua,Uub)) ).

% ATP.lambda_848
tff(fact_9029_ATP_Olambda__849,axiom,
    ! [D: $tType,F2: $tType,B: $tType,Uu: fun(F2,fun(D,bool)),Uua: fun(B,F2),Uub: B] : aa(B,fun(D,bool),aa(fun(B,F2),fun(B,fun(D,bool)),aTP_Lamp_auv(fun(F2,fun(D,bool)),fun(fun(B,F2),fun(B,fun(D,bool))),Uu),Uua),Uub) = aa(F2,fun(D,bool),Uu,aa(B,F2,Uua,Uub)) ).

% ATP.lambda_849
tff(fact_9030_ATP_Olambda__850,axiom,
    ! [C: $tType,E: $tType,A: $tType,Uu: fun(E,fun(C,bool)),Uua: fun(A,E),Uub: A] : aa(A,fun(C,bool),aa(fun(A,E),fun(A,fun(C,bool)),aTP_Lamp_auu(fun(E,fun(C,bool)),fun(fun(A,E),fun(A,fun(C,bool))),Uu),Uua),Uub) = aa(E,fun(C,bool),Uu,aa(A,E,Uua,Uub)) ).

% ATP.lambda_850
tff(fact_9031_ATP_Olambda__851,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_abg(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_851
tff(fact_9032_ATP_Olambda__852,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_zy(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_852
tff(fact_9033_ATP_Olambda__853,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_ar(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).

% ATP.lambda_853
tff(fact_9034_ATP_Olambda__854,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_vk(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_854
tff(fact_9035_ATP_Olambda__855,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ars(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_855
tff(fact_9036_ATP_Olambda__856,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_afn(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_856
tff(fact_9037_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Uu: fun(A,B),Uua: fun(B,A),Uub: B] : aa(B,B,aa(fun(B,A),fun(B,B),aTP_Lamp_agd(fun(A,B),fun(fun(B,A),fun(B,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_857
tff(fact_9038_ATP_Olambda__858,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aqc(fun(A,bool),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uu,aa(B,A,Uua,Uub))) ) ).

% ATP.lambda_858
tff(fact_9039_ATP_Olambda__859,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,bool),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_ajp(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),Uu),Uua),Uub) = aa(A,fun(B,bool),Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_859
tff(fact_9040_ATP_Olambda__860,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: D] : aa(D,fun(B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_ov(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),Uu),Uua),Uub) = aa(A,fun(B,C),Uu,aa(D,A,Uua,Uub)) ).

% ATP.lambda_860
tff(fact_9041_ATP_Olambda__861,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(num,A),Uub: num] : aa(num,B,aa(fun(num,A),fun(num,B),aTP_Lamp_fi(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_861
tff(fact_9042_ATP_Olambda__862,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_fh(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).

% ATP.lambda_862
tff(fact_9043_ATP_Olambda__863,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aob(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_863
tff(fact_9044_ATP_Olambda__864,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,C),Uua: fun(C,bool),Uub: D] :
      ( pp(aa(D,bool,aa(fun(C,bool),fun(D,bool),aTP_Lamp_aum(fun(D,C),fun(fun(C,bool),fun(D,bool)),Uu),Uua),Uub))
    <=> pp(aa(C,bool,Uua,aa(D,C,Uu,Uub))) ) ).

% ATP.lambda_864
tff(fact_9045_ATP_Olambda__865,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,D),Uua: fun(D,bool),Uub: C] :
      ( pp(aa(C,bool,aa(fun(D,bool),fun(C,bool),aTP_Lamp_auo(fun(C,D),fun(fun(D,bool),fun(C,bool)),Uu),Uua),Uub))
    <=> pp(aa(D,bool,Uua,aa(C,D,Uu,Uub))) ) ).

% ATP.lambda_865
tff(fact_9046_ATP_Olambda__866,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_aok(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_866
tff(fact_9047_ATP_Olambda__867,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_aol(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_867
tff(fact_9048_ATP_Olambda__868,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_aaf(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_868
tff(fact_9049_ATP_Olambda__869,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(B,A),Uub: A] : aa(A,A,aa(fun(B,A),fun(A,A),aTP_Lamp_agc(fun(A,B),fun(fun(B,A),fun(A,A)),Uu),Uua),Uub) = aa(B,A,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_869
tff(fact_9050_ATP_Olambda__870,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_afe(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uua,aa(A,B,Uu,Uub))) ) ).

% ATP.lambda_870
tff(fact_9051_ATP_Olambda__871,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_wb(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_871
tff(fact_9052_ATP_Olambda__872,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqx(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ).

% ATP.lambda_872
tff(fact_9053_ATP_Olambda__873,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_abw(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).

% ATP.lambda_873
tff(fact_9054_ATP_Olambda__874,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_po(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).

% ATP.lambda_874
tff(fact_9055_ATP_Olambda__875,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_atj(fun(B,bool),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_875
tff(fact_9056_ATP_Olambda__876,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ot(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).

% ATP.lambda_876
tff(fact_9057_ATP_Olambda__877,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_ahl(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_877
tff(fact_9058_ATP_Olambda__878,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,heap_Time_Heap(A)),Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_br(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(B,heap_Time_Heap(A),Uua,Uu) ).

% ATP.lambda_878
tff(fact_9059_ATP_Olambda__879,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_ajb(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_aja(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).

% ATP.lambda_879
tff(fact_9060_ATP_Olambda__880,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: A] : aa(A,option(A),aa(A,fun(A,option(A)),aTP_Lamp_ba(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Uu,Uua),Uub)) ).

% ATP.lambda_880
tff(fact_9061_ATP_Olambda__881,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(B),Uub: C] : aa(C,A,aa(multiset(B),fun(C,A),aTP_Lamp_aiv(fun(B,fun(C,A)),fun(multiset(B),fun(C,A)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(A,aa(multiset(B),multiset(A),image_mset(B,A,aa(C,fun(B,A),aTP_Lamp_fx(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_881
tff(fact_9062_ATP_Olambda__882,axiom,
    ! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: product_prod(B,C)] : aa(product_prod(B,C),set(A),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_zt(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),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(D,E)),set(set(A)),image2(product_prod(D,E),set(A),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_zs(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),set(A))),Uu),Uub)),Uua)) ).

% ATP.lambda_882
tff(fact_9063_ATP_Olambda__883,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_xa(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),image2(B,A,aa(C,fun(B,A),aTP_Lamp_wz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_883
tff(fact_9064_ATP_Olambda__884,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_ym(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(C,fun(B,A),aTP_Lamp_wz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_884
tff(fact_9065_ATP_Olambda__885,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: multiset(C),Uub: B] : aa(B,A,aa(multiset(C),fun(B,A),aTP_Lamp_aiu(fun(B,fun(C,A)),fun(multiset(C),fun(B,A)),Uu),Uua),Uub) = comm_m7189776963980413722m_mset(A,aa(multiset(C),multiset(A),image_mset(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_885
tff(fact_9066_ATP_Olambda__886,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_wy(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),image2(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_886
tff(fact_9067_ATP_Olambda__887,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,set(C))),Uua: set(B),Uub: A] : aa(A,set(C),aa(set(B),fun(A,set(C)),aTP_Lamp_aka(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),Uu),Uua),Uub) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uu,Uub)),Uua)) ).

% ATP.lambda_887
tff(fact_9068_ATP_Olambda__888,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_yl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_888
tff(fact_9069_ATP_Olambda__889,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ue(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_889
tff(fact_9070_ATP_Olambda__890,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ug(B,fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_890
tff(fact_9071_ATP_Olambda__891,axiom,
    ! [S8: $tType,R9: $tType,Q8: $tType,Uu: fun(R9,set(S8)),Uua: fun(Q8,set(R9)),Uub: Q8] : aa(Q8,set(S8),aa(fun(Q8,set(R9)),fun(Q8,set(S8)),aTP_Lamp_wx(fun(R9,set(S8)),fun(fun(Q8,set(R9)),fun(Q8,set(S8))),Uu),Uua),Uub) = aa(set(set(S8)),set(S8),complete_Sup_Sup(set(S8)),aa(set(R9),set(set(S8)),image2(R9,set(S8),Uu),aa(Q8,set(R9),Uua,Uub))) ).

% ATP.lambda_891
tff(fact_9072_ATP_Olambda__892,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_xc(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)),image2(B,set(A),Uu),aa(C,set(B),Uua,Uub))) ).

% ATP.lambda_892
tff(fact_9073_ATP_Olambda__893,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_xb(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),image2(B,A,Uu),aa(C,set(B),Uua,Uub))) ) ).

% ATP.lambda_893
tff(fact_9074_ATP_Olambda__894,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_aro(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).

% ATP.lambda_894
tff(fact_9075_ATP_Olambda__895,axiom,
    ! [S8: $tType,R9: $tType,Q8: $tType,Uu: fun(R9,set(S8)),Uua: fun(Q8,set(R9)),Uub: Q8] : aa(Q8,set(S8),aa(fun(Q8,set(R9)),fun(Q8,set(S8)),aTP_Lamp_zd(fun(R9,set(S8)),fun(fun(Q8,set(R9)),fun(Q8,set(S8))),Uu),Uua),Uub) = aa(set(set(S8)),set(S8),complete_Inf_Inf(set(S8)),aa(set(R9),set(set(S8)),image2(R9,set(S8),Uu),aa(Q8,set(R9),Uua,Uub))) ).

% ATP.lambda_895
tff(fact_9076_ATP_Olambda__896,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_asr(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).

% ATP.lambda_896
tff(fact_9077_ATP_Olambda__897,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_lo(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_897
tff(fact_9078_ATP_Olambda__898,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_aby(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),Uu),Uua)) ).

% ATP.lambda_898
tff(fact_9079_ATP_Olambda__899,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(A,fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_bw(fun(A,A),fun(A,fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_899
tff(fact_9080_ATP_Olambda__900,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: list(B)] : aa(list(B),fun(list(A),list(A)),aa(B,fun(list(B),fun(list(A),list(A))),aTP_Lamp_aoa(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),Uu),Uua),Uub) = aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_900
tff(fact_9081_ATP_Olambda__901,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_mi(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_mh(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub))) ).

% ATP.lambda_901
tff(fact_9082_ATP_Olambda__902,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
      ( pp(aa(A,bool,aa(set(C),fun(A,bool),aTP_Lamp_ada(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [B7: B] : pp(aa(set(C),bool,member(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B7)),Uua)) ) ).

% ATP.lambda_902
tff(fact_9083_ATP_Olambda__903,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_aes(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [I: nat] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I)),aa(nat,B,nth(B,Uua),I)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua)))) ) ) ).

% ATP.lambda_903
tff(fact_9084_ATP_Olambda__904,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_aew(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [A9: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),aa(A,B,Uua,A9)) )
          & pp(aa(set(A),bool,member(A,A9),Uu)) ) ) ).

% ATP.lambda_904
tff(fact_9085_ATP_Olambda__905,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_adz(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Uu)))
          & pp(aa(set(nat),bool,member(nat,I),Uua)) ) ) ).

% ATP.lambda_905
tff(fact_9086_ATP_Olambda__906,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_aea(nat,fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uua),I) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uu),I))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Uua))) ) ) ).

% ATP.lambda_906
tff(fact_9087_ATP_Olambda__907,axiom,
    ! [A: $tType,Uu: set(set(product_prod(A,A))),Uua: set(product_prod(A,A)),Uub: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_auk(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua),Uub))
    <=> ? [R4: set(product_prod(A,A))] :
          ( ( Uub = 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))),minus_minus(set(product_prod(A,A))),R4),Uua) )
          & pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),R4),Uu)) ) ) ).

% ATP.lambda_907
tff(fact_9088_ATP_Olambda__908,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_agz(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A9: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A9) )
              & pp(aa(set(A),bool,member(A,A9),Uu)) ) ) ) ).

% ATP.lambda_908
tff(fact_9089_ATP_Olambda__909,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_adh(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B7) )
              & pp(aa(set(A),bool,member(A,B7),Uua)) ) ) ) ).

% ATP.lambda_909
tff(fact_9090_ATP_Olambda__910,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_agp(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A9: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A9) )
              & pp(aa(set(A),bool,member(A,A9),Uu)) ) ) ) ).

% ATP.lambda_910
tff(fact_9091_ATP_Olambda__911,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_auc(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A9: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),Uua),A9) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A9),Uu)) ) ) ).

% ATP.lambda_911
tff(fact_9092_ATP_Olambda__912,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: multiset(A),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_aue(set(multiset(A)),fun(multiset(A),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A9: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),Uua),A9) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A9),Uu)) ) ) ).

% ATP.lambda_912
tff(fact_9093_ATP_Olambda__913,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_aeh(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [I: nat] :
          ( ( Uub = aa(B,set(A),Uu,aa(nat,B,nth(B,Uua),I)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_913
tff(fact_9094_ATP_Olambda__914,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_aeo(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [A9: B] :
          ( ( Uub = aa(B,set(A),Uu,A9) )
          & pp(aa(set(B),bool,member(B,A9),aa(list(B),set(B),set2(B),Uua))) ) ) ).

% ATP.lambda_914
tff(fact_9095_ATP_Olambda__915,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_acw(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [L2: B] :
          ( ( Uub = aa(B,A,Uu,L2) )
          & pp(aa(set(B),bool,member(B,L2),Uua)) ) ) ).

% ATP.lambda_915
tff(fact_9096_ATP_Olambda__916,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: fun(C,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(C,bool),fun(B,bool),aTP_Lamp_ame(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X4: C] :
          ( pp(aa(set(C),bool,member(C,X4),aa(B,set(C),Uu,Uub)))
         => pp(aa(C,bool,Uua,X4)) ) ) ).

% ATP.lambda_916
tff(fact_9097_ATP_Olambda__917,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,bool),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(fun(B,bool),fun(set(A),bool),aTP_Lamp_ade(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( ( Uub = aa(B,set(A),Uu,X4) )
          & pp(aa(B,bool,Uua,X4)) ) ) ).

% ATP.lambda_917
tff(fact_9098_ATP_Olambda__918,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_acy(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( ( Uub = aa(B,A,Uu,X4) )
          & pp(aa(B,bool,Uua,X4)) ) ) ).

% ATP.lambda_918
tff(fact_9099_ATP_Olambda__919,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,B),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aTP_Lamp_add(fun(A,bool),fun(fun(A,B),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( ( Uub = aa(A,B,Uua,X4) )
          & pp(aa(A,bool,Uu,X4)) ) ) ).

% ATP.lambda_919
tff(fact_9100_ATP_Olambda__920,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,fun(A,bool)),Uub: C] :
      ( pp(aa(C,bool,aa(fun(C,fun(A,bool)),fun(C,bool),aTP_Lamp_apj(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(C,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))))
          & pp(aa(A,bool,aa(C,fun(A,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_920
tff(fact_9101_ATP_Olambda__921,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,C),Uub: fun(A,C)] :
      ( pp(aa(fun(A,C),bool,aa(fun(A,C),fun(fun(A,C),bool),aTP_Lamp_apl(fun(A,fun(B,bool)),fun(fun(A,C),fun(fun(A,C),bool)),Uu),Uua),Uub))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))))
         => ( aa(A,C,Uua,X4) = aa(A,C,Uub,X4) ) ) ) ).

% ATP.lambda_921
tff(fact_9102_ATP_Olambda__922,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
      ( pp(aa(A,bool,aa(list(B),fun(A,bool),aTP_Lamp_adv(fun(B,option(A)),fun(list(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),aa(list(B),set(B),set2(B),Uua)))
          & ( aa(B,option(A),Uu,X4) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_922
tff(fact_9103_ATP_Olambda__923,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_amb(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uu))
         => pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_923
tff(fact_9104_ATP_Olambda__924,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aqb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uu))
         => pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_924
tff(fact_9105_ATP_Olambda__925,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_alu(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uu))
          & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_925
tff(fact_9106_ATP_Olambda__926,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_all(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uu))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X4)) ) ) ).

% ATP.lambda_926
tff(fact_9107_ATP_Olambda__927,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_alm(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uua))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uu,X4),Uub)) ) ) ).

% ATP.lambda_927
tff(fact_9108_ATP_Olambda__928,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_amm(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X4),Uu))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_aml(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)),X4)) ) ) ).

% ATP.lambda_928
tff(fact_9109_ATP_Olambda__929,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_acz(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [P6: product_prod(A,B)] :
          ( ( Uub = P6 )
          & pp(aa(set(C),bool,member(C,aa(B,C,aa(A,fun(B,C),Uu,aa(product_prod(A,B),A,product_fst(A,B),P6)),aa(product_prod(A,B),B,product_snd(A,B),P6))),Uua)) ) ) ).

% ATP.lambda_929
tff(fact_9110_ATP_Olambda__930,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_alp(set(product_prod(A,B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Uub)),Uu)) ) ) ).

% ATP.lambda_930
tff(fact_9111_ATP_Olambda__931,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_alr(fun(A,B),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & ( aa(A,B,Uu,X4) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_931
tff(fact_9112_ATP_Olambda__932,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_alx(fun(A,option(B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & ( aa(A,option(B),Uu,X4) = aa(B,option(B),some(B),Uub) ) ) ) ).

% ATP.lambda_932
tff(fact_9113_ATP_Olambda__933,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_amc(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uua))
         => pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X4))) ) ) ).

% ATP.lambda_933
tff(fact_9114_ATP_Olambda__934,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_alt(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),Uua))
          & pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X4))) ) ) ).

% ATP.lambda_934
tff(fact_9115_ATP_Olambda__935,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_als(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),Uua))
          & ( Uub = aa(A,B,Uu,X4) ) ) ) ).

% ATP.lambda_935
tff(fact_9116_ATP_Olambda__936,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: filter(B),Uub: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(filter(B),fun(fun(A,bool),bool),aTP_Lamp_aqw(fun(A,B),fun(filter(B),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ? [Q9: fun(B,bool)] :
          ( eventually(B,Q9,Uua)
          & ! [X4: A] :
              ( pp(aa(B,bool,Q9,aa(A,B,Uu,X4)))
             => pp(aa(A,bool,Uub,X4)) ) ) ) ).

% ATP.lambda_936
tff(fact_9117_ATP_Olambda__937,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_act(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [Y4: A] :
          ( pp(aa(A,bool,Uu,Y4))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),Y4)) ) ) ).

% ATP.lambda_937
tff(fact_9118_ATP_Olambda__938,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_adf(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(B,bool,Uua,X4))
          & pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X4))) ) ) ).

% ATP.lambda_938
tff(fact_9119_ATP_Olambda__939,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ahu(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N3: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N3),Uu),Uua) ) ).

% ATP.lambda_939
tff(fact_9120_ATP_Olambda__940,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,bool)),Uub: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_acx(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A9: B,B7: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A9)),aa(B,A,Uu,B7)) )
          & pp(aa(B,bool,aa(B,fun(B,bool),Uua,A9),B7)) ) ) ).

% ATP.lambda_940
tff(fact_9121_ATP_Olambda__941,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_adm(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A19: B,A25: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A19)),aa(B,A,Uua,A25)) )
          & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A19),A25)),Uu)) ) ) ).

% ATP.lambda_941
tff(fact_9122_ATP_Olambda__942,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_aeg(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [A9: A,V5: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A9),V5)) )
          | ? [U7: list(A),Aa3: A,B7: A,Va3: list(A),W3: list(A)] :
              ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B7)),Uu))
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U7),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa3),Va3)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U7),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),W3)) ) ) ) ) ).

% ATP.lambda_942
tff(fact_9123_ATP_Olambda__943,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_agy(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A9: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A9),B7) )
              & pp(aa(set(A),bool,member(A,A9),Uu))
              & pp(aa(set(A),bool,member(A,B7),Uua)) ) ) ) ).

% ATP.lambda_943
tff(fact_9124_ATP_Olambda__944,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_agq(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A9: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A9),B7) )
              & pp(aa(set(A),bool,member(A,A9),Uu))
              & pp(aa(set(A),bool,member(A,B7),Uua)) ) ) ) ).

% ATP.lambda_944
tff(fact_9125_ATP_Olambda__945,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_aub(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A9: multiset(A),B7: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),union_mset(A),A9),B7) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A9),Uu))
          & pp(aa(set(multiset(A)),bool,member(multiset(A),B7),Uua)) ) ) ).

% ATP.lambda_945
tff(fact_9126_ATP_Olambda__946,axiom,
    ! [A: $tType,Uu: set(multiset(A)),Uua: set(multiset(A)),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(set(multiset(A)),fun(multiset(A),bool),aTP_Lamp_auf(set(multiset(A)),fun(set(multiset(A)),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A9: multiset(A),B7: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),inter_mset(A),A9),B7) )
          & pp(aa(set(multiset(A)),bool,member(multiset(A),A9),Uu))
          & pp(aa(set(multiset(A)),bool,member(multiset(A),B7),Uua)) ) ) ).

% ATP.lambda_946
tff(fact_9127_ATP_Olambda__947,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(list(A)),fun(list(A),bool),aTP_Lamp_adb(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [X4: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
          & pp(aa(set(A),bool,member(A,X4),Uu))
          & pp(aa(set(list(A)),bool,member(list(A),Xs3),Uua)) ) ) ).

% ATP.lambda_947
tff(fact_9128_ATP_Olambda__948,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_aqt(filter(A),fun(filter(A),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ? [Q9: fun(A,bool),R13: fun(A,bool)] :
          ( eventually(A,Q9,Uu)
          & eventually(A,R13,Uua)
          & ! [X4: A] :
              ( ( pp(aa(A,bool,Q9,X4))
                & pp(aa(A,bool,R13,X4)) )
             => pp(aa(A,bool,Uub,X4)) ) ) ) ).

% ATP.lambda_948
tff(fact_9129_ATP_Olambda__949,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: multiset(A),Uub: multiset(A)] :
      ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_alc(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),bool)),Uu),Uua),Uub))
    <=> ? [A9: A,M03: multiset(A),K9: multiset(A)] :
          ( ( Uub = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A9),M03) )
          & ( Uua = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M03),K9) )
          & ! [B7: A] :
              ( pp(aa(set(A),bool,member(A,B7),aa(multiset(A),set(A),set_mset(A),K9)))
             => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),A9)),Uu)) ) ) ) ).

% ATP.lambda_949
tff(fact_9130_ATP_Olambda__950,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_adl(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [Us3: list(A),Z2: A,Z13: A,Vs3: list(A)] :
          ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Vs3)) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Z13)),Uu))
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z13),Vs3)) ) ) ) ).

% ATP.lambda_950
tff(fact_9131_ATP_Olambda__951,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_dp(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_951
tff(fact_9132_ATP_Olambda__952,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ul(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(product_prod(C,A),A,product_snd(C,A),Uu)),Uua),aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_952
tff(fact_9133_ATP_Olambda__953,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_aig(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_953
tff(fact_9134_ATP_Olambda__954,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_ahz(set(A),fun(fun(A,B),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uua),Uu)),the_inv_into(A,B,Uu,Uua,Uuc),Uub) ).

% ATP.lambda_954
tff(fact_9135_ATP_Olambda__955,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: A,Uub: B,Uuc: set(B)] : aa(set(B),set(B),aa(B,fun(set(B),set(B)),aa(A,fun(B,fun(set(B),set(B))),aTP_Lamp_va(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = if(set(B),aa(set(A),bool,member(A,Uua),Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uub),Uuc),Uuc) ).

% ATP.lambda_955
tff(fact_9136_ATP_Olambda__956,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_hl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_956
tff(fact_9137_ATP_Olambda__957,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_ji(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_957
tff(fact_9138_ATP_Olambda__958,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_jj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_958
tff(fact_9139_ATP_Olambda__959,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_hm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_959
tff(fact_9140_ATP_Olambda__960,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_ahq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,member(A,Uuc),Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_960
tff(fact_9141_ATP_Olambda__961,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,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_lp(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_961
tff(fact_9142_ATP_Olambda__962,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,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_ll(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_962
tff(fact_9143_ATP_Olambda__963,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ms(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).

% ATP.lambda_963
tff(fact_9144_ATP_Olambda__964,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B,Uub: fun(A,option(B)),Uuc: A] : aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_aia(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uu),aa(B,option(B),some(B),Uua),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_964
tff(fact_9145_ATP_Olambda__965,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B,Uuc: A] : aa(A,option(B),aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_nd(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uua),aa(B,option(B),some(B),Uub),aa(A,option(B),Uu,Uuc)) ).

% ATP.lambda_965
tff(fact_9146_ATP_Olambda__966,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: B,Uub: set(A),Uuc: A] : aa(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_aff(B,fun(B,fun(set(A),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,member(A,Uuc),Uub),Uu,Uua) ).

% ATP.lambda_966
tff(fact_9147_ATP_Olambda__967,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A),Uuc: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_aty(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = if(product_prod(list(A),list(A)),aa(A,bool,Uu,Uua),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),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)),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uuc))) ).

% ATP.lambda_967
tff(fact_9148_ATP_Olambda__968,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,bool),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_lk(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_968
tff(fact_9149_ATP_Olambda__969,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,bool),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_lj(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_969
tff(fact_9150_ATP_Olambda__970,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),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_vf(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_970
tff(fact_9151_ATP_Olambda__971,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,option(B)),Uub: fun(A,option(B)),Uuc: A] : aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_aid(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,Uu,Uuc),aa(A,option(B),Uua,Uuc),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_971
tff(fact_9152_ATP_Olambda__972,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,bool),fun(fun(A,B),fun(A,B)),aTP_Lamp_aoc(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_972
tff(fact_9153_ATP_Olambda__973,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ut(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_us(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_973
tff(fact_9154_ATP_Olambda__974,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_uo(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_un(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_974
tff(fact_9155_ATP_Olambda__975,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(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_mv(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uuc),aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_975
tff(fact_9156_ATP_Olambda__976,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,fun(list(B),fun(C,C))),Uua: fun(A,B),Uub: A,Uuc: list(A)] : aa(list(A),fun(C,C),aa(A,fun(list(A),fun(C,C)),aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_aof(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Uu),Uua),Uub),Uuc) = aa(list(B),fun(C,C),aa(B,fun(list(B),fun(C,C)),Uu,aa(A,B,Uua,Uub)),aa(list(A),list(B),map(A,B,Uua),Uuc)) ).

% ATP.lambda_976
tff(fact_9157_ATP_Olambda__977,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_rt(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ).

% ATP.lambda_977
tff(fact_9158_ATP_Olambda__978,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_rs(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).

% ATP.lambda_978
tff(fact_9159_ATP_Olambda__979,axiom,
    ! [B: $tType,F2: $tType,D: $tType,Uu: fun(B,fun(F2,bool)),Uua: fun(D,F2),Uub: B,Uuc: D] :
      ( pp(aa(D,bool,aa(B,fun(D,bool),aa(fun(D,F2),fun(B,fun(D,bool)),aTP_Lamp_aux(fun(B,fun(F2,bool)),fun(fun(D,F2),fun(B,fun(D,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(F2,bool,aa(B,fun(F2,bool),Uu,Uub),aa(D,F2,Uua,Uuc))) ) ).

% ATP.lambda_979
tff(fact_9160_ATP_Olambda__980,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_afy(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_980
tff(fact_9161_ATP_Olambda__981,axiom,
    ! [A: $tType,E: $tType,C: $tType,Uu: fun(A,fun(E,bool)),Uua: fun(C,E),Uub: A,Uuc: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,E),fun(A,fun(C,bool)),aTP_Lamp_auw(fun(A,fun(E,bool)),fun(fun(C,E),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(E,bool,aa(A,fun(E,bool),Uu,Uub),aa(C,E,Uua,Uuc))) ) ).

% ATP.lambda_981
tff(fact_9162_ATP_Olambda__982,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_abf(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).

% ATP.lambda_982
tff(fact_9163_ATP_Olambda__983,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,B),Uub: A,Uuc: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_ajo(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(C,B,Uua,Uuc))) ) ).

% ATP.lambda_983
tff(fact_9164_ATP_Olambda__984,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_ru(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_984
tff(fact_9165_ATP_Olambda__985,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_gb(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),Uu),Uua),Uub),Uuc) = groups7311177749621191930dd_sum(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ga(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc),Uub) ) ).

% ATP.lambda_985
tff(fact_9166_ATP_Olambda__986,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: filter(A),Uuc: fun(B,bool)] :
      ( pp(aa(fun(B,bool),bool,aa(filter(A),fun(fun(B,bool),bool),aa(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),aTP_Lamp_aqs(set(A),fun(fun(A,B),fun(filter(A),fun(fun(B,bool),bool))),Uu),Uua),Uub),Uuc))
    <=> eventually(A,aa(fun(B,bool),fun(A,bool),aa(fun(A,B),fun(fun(B,bool),fun(A,bool)),aTP_Lamp_aqr(set(A),fun(fun(A,B),fun(fun(B,bool),fun(A,bool))),Uu),Uua),Uuc),Uub) ) ).

% ATP.lambda_986
tff(fact_9167_ATP_Olambda__987,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B,Uuc: B] : aa(B,fun(list(B),list(B)),aa(B,fun(B,fun(list(B),list(B))),aa(list(B),fun(B,fun(B,fun(list(B),list(B)))),aTP_Lamp_px(fun(B,A),fun(list(B),fun(B,fun(B,fun(list(B),list(B))))),Uu),Uua),Uub),Uuc) = case_list(list(B),B,if(list(B),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uu,Uuc)),Uua,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uuc),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),nil(B)))),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_pw(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua)) ) ).

% ATP.lambda_987
tff(fact_9168_ATP_Olambda__988,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,fun(A,bool)),Uub: set(A),Uuc: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_aul(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,pred_chain(A,Uu,Uua),Uuc))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uuc)) ) ) ).

% ATP.lambda_988
tff(fact_9169_ATP_Olambda__989,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: bool,Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa(bool,fun(A,fun(B,assn)),aTP_Lamp_co(fun(A,fun(B,assn)),fun(bool,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_989
tff(fact_9170_ATP_Olambda__990,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(D,E),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(product_prod(D,E),fun(B,fun(C,set(A))),aTP_Lamp_zr(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(D,E),set(A),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc)),Uua) ).

% ATP.lambda_990
tff(fact_9171_ATP_Olambda__991,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_kq(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_ih(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_kn(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_991
tff(fact_9172_ATP_Olambda__992,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: C] : aa(C,A,aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aTP_Lamp_ko(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_fx(fun(B,fun(C,A)),fun(C,fun(B,A)),Uua),Uuc),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_kn(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_992
tff(fact_9173_ATP_Olambda__993,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_ne(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uu),Uua),Uub),Uuc) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_Pair(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),Uu)),Uuc)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Uub),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one))))))))))))))))))))))))))))))),Uua))) ).

% ATP.lambda_993
tff(fact_9174_ATP_Olambda__994,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: set(A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_akf(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),product_Sigma(A,A,Uua,aTP_Lamp_abo(set(A),fun(A,set(A)),Uua))))
        & pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uuc)) ) ) ).

% ATP.lambda_994
tff(fact_9175_ATP_Olambda__995,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_nm(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = 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),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).

% ATP.lambda_995
tff(fact_9176_ATP_Olambda__996,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aot(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))) ) ).

% ATP.lambda_996
tff(fact_9177_ATP_Olambda__997,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_aez(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uu))
        & ~ pp(aa(set(A),bool,member(A,Uub),Uua))
        & ~ pp(aa(set(A),bool,member(A,Uuc),Uua)) ) ) ).

% ATP.lambda_997
tff(fact_9178_ATP_Olambda__998,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_el(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uua)) ) ) ) ).

% ATP.lambda_998
tff(fact_9179_ATP_Olambda__999,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_ajh(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc))),Uu)) ) ).

% ATP.lambda_999
tff(fact_9180_ATP_Olambda__1000,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),bool),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_ek(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
        & pp(aa(product_prod(A,B),bool,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc))) ) ) ).

% ATP.lambda_1000
tff(fact_9181_ATP_Olambda__1001,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_age(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uuc)),Uua)),Uub) ) ).

% ATP.lambda_1001
tff(fact_9182_ATP_Olambda__1002,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A),Uuc: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_ard(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),Uu),Uua),Uub),Uuc) = 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)),sup_sup(set(A)),aa(set(A),set(A),Uua,Uuc)),Uub)),Uu) ).

% ATP.lambda_1002
tff(fact_9183_ATP_Olambda__1003,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_hj(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_1003
tff(fact_9184_ATP_Olambda__1004,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_hf(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,power_power(nat,Uu),Uuc))),aa(nat,nat,power_power(nat,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_1004
tff(fact_9185_ATP_Olambda__1005,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_hi(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,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_1005
tff(fact_9186_ATP_Olambda__1006,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_aef(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X4: A,Y4: A,Xs6: list(A),Ys6: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4)),Uu)) ) ) ) ).

% ATP.lambda_1006
tff(fact_9187_ATP_Olambda__1007,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_jw(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,power_power(A,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,power_power(A,Uu),Uuc)) ) ).

% ATP.lambda_1007
tff(fact_9188_ATP_Olambda__1008,axiom,
    ! [A: $tType,Uu: bool,Uua: A,Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_pd(bool,fun(A,fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( pp(Uu)
         => ( Uuc = Uua ) )
        & ( ~ pp(Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_1008
tff(fact_9189_ATP_Olambda__1009,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_vl(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),aa(set(B),set(A),image2(B,A,Uu),Uua)))
        & pp(aa(A,bool,Uub,Uuc)) ) ) ).

% ATP.lambda_1009
tff(fact_9190_ATP_Olambda__1010,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: multiset(B),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(multiset(B),fun(A,fun(B,bool)),aTP_Lamp_akk(fun(B,A),fun(multiset(B),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),aa(multiset(B),set(B),set_mset(B),Uua)))
        & ( Uub = aa(B,A,Uu,Uuc) ) ) ) ).

% ATP.lambda_1010
tff(fact_9191_ATP_Olambda__1011,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: fun(B,fun(C,bool)),Uub: B,Uuc: C] :
      ( pp(aa(C,bool,aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_kl(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(C),bool,member(C,Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_1011
tff(fact_9192_ATP_Olambda__1012,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_mh(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_1012
tff(fact_9193_ATP_Olambda__1013,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,fun(C,bool)),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,fun(C,bool)),fun(C,fun(B,bool)),aTP_Lamp_kn(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
        & pp(aa(C,bool,aa(B,fun(C,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_1013
tff(fact_9194_ATP_Olambda__1014,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,bool)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_ly(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_1014
tff(fact_9195_ATP_Olambda__1015,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(product_prod(B,B)),fun(B,fun(A,bool)),aTP_Lamp_asw(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & ( aa(A,B,bNF_Greatest_toCard(A,B,Uu,Uua),Uuc) = Uub ) ) ) ).

% ATP.lambda_1015
tff(fact_9196_ATP_Olambda__1016,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_vn(set(A),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_1016
tff(fact_9197_ATP_Olambda__1017,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_ato(fun(A,B),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uua))
        & ( aa(A,B,Uu,Uub) = aa(A,B,Uu,Uuc) ) ) ) ).

% ATP.lambda_1017
tff(fact_9198_ATP_Olambda__1018,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_vp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
        & ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1018
tff(fact_9199_ATP_Olambda__1019,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_wc(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1019
tff(fact_9200_ATP_Olambda__1020,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_ahw(fun(A,B),fun(set(A),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uua))
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_1020
tff(fact_9201_ATP_Olambda__1021,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( pp(aa(B,bool,aa(set(B),fun(B,bool),aa(fun(B,A),fun(set(B),fun(B,bool)),aTP_Lamp_ahk(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uub))
        & pp(aa(set(A),bool,member(A,aa(B,A,Uua,Uuc)),Uu)) ) ) ).

% ATP.lambda_1021
tff(fact_9202_ATP_Olambda__1022,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_jx(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,power_power(A,Uu),Uuc)),aa(nat,A,power_power(A,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).

% ATP.lambda_1022
tff(fact_9203_ATP_Olambda__1023,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_hc(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_1023
tff(fact_9204_ATP_Olambda__1024,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: list(B),Uuc: list(B)] : aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),aa(list(B),fun(list(B),fun(list(B),list(B))),aTP_Lamp_pu(fun(B,A),fun(list(B),fun(list(B),fun(list(B),list(B)))),Uu),Uua),Uub),Uuc) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),aa(list(B),list(B),linorder_sort_key(B,A,Uu),Uua)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uub),aa(list(B),list(B),linorder_sort_key(B,A,Uu),Uuc))) ) ).

% ATP.lambda_1024
tff(fact_9205_ATP_Olambda__1025,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_oe(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_1025
tff(fact_9206_ATP_Olambda__1026,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_nt(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_1026
tff(fact_9207_ATP_Olambda__1027,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_og(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_1027
tff(fact_9208_ATP_Olambda__1028,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_nr(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_1028
tff(fact_9209_ATP_Olambda__1029,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_nv(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = 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),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).

% ATP.lambda_1029
tff(fact_9210_ATP_Olambda__1030,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_nx(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = 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),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).

% ATP.lambda_1030
tff(fact_9211_ATP_Olambda__1031,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_qb(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)),aa(product_prod(A,B),fun(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_1031
tff(fact_9212_ATP_Olambda__1032,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_qc(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)),aa(product_prod(A,B),fun(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_1032
tff(fact_9213_ATP_Olambda__1033,axiom,
    ! [A: $tType,B: $tType,Uu: filter(A),Uua: filter(B),Uub: fun(A,bool),Uuc: fun(B,bool)] :
      ( pp(aa(fun(B,bool),bool,aa(fun(A,bool),fun(fun(B,bool),bool),aa(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool)),aTP_Lamp_atg(filter(A),fun(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool))),Uu),Uua),Uub),Uuc))
    <=> ( eventually(A,Uub,Uu)
        & eventually(B,Uuc,Uua) ) ) ).

% ATP.lambda_1033
tff(fact_9214_ATP_Olambda__1034,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_pa(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_1034
tff(fact_9215_ATP_Olambda__1035,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_afk(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua) ) ) ).

% ATP.lambda_1035
tff(fact_9216_ATP_Olambda__1036,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_vm(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uua))
        & pp(aa(A,bool,Uub,aa(B,A,Uu,Uuc))) ) ) ).

% ATP.lambda_1036
tff(fact_9217_ATP_Olambda__1037,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),aTP_Lamp_zk(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(set(product_prod(A,B)),bool,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_1037
tff(fact_9218_ATP_Olambda__1038,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ku(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).

% ATP.lambda_1038
tff(fact_9219_ATP_Olambda__1039,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ks(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_1039
tff(fact_9220_ATP_Olambda__1040,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_dk(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),snga_assn(A,Uub,Uua)),pure_assn(aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),aa(nat,A,nth(A,Uua),Uu)))) ) ).

% ATP.lambda_1040
tff(fact_9221_ATP_Olambda__1041,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_jh(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,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_1041
tff(fact_9222_ATP_Olambda__1042,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,bool),Uub: fun(A,B),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),aTP_Lamp_aqn(set(A),fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(B,bool,Uua,aa(A,B,Uub,Uuc)))
        & pp(aa(set(A),bool,member(A,Uuc),Uu)) ) ) ).

% ATP.lambda_1042
tff(fact_9223_ATP_Olambda__1043,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: fun(B,bool),Uuc: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aa(fun(A,B),fun(fun(B,bool),fun(A,bool)),aTP_Lamp_aqr(set(A),fun(fun(A,B),fun(fun(B,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(B,bool,Uub,aa(A,B,Uua,Uuc)))
        & pp(aa(set(A),bool,member(A,Uuc),Uu)) ) ) ).

% ATP.lambda_1043
tff(fact_9224_ATP_Olambda__1044,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_abh(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_1044
tff(fact_9225_ATP_Olambda__1045,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_abx(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_abw(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_1045
tff(fact_9226_ATP_Olambda__1046,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_ajd(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uuc))
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_1046
tff(fact_9227_ATP_Olambda__1047,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_rd(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_1047
tff(fact_9228_ATP_Olambda__1048,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( semiring_0(B)
        & comm_monoid_add(A)
        & times(A) )
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_aiw(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_1048
tff(fact_9229_ATP_Olambda__1049,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_aim(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).

% ATP.lambda_1049
tff(fact_9230_ATP_Olambda__1050,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ga(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_1050
tff(fact_9231_ATP_Olambda__1051,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_zb(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_1051
tff(fact_9232_ATP_Olambda__1052,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_yp(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_1052
tff(fact_9233_ATP_Olambda__1053,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_xm(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_1053
tff(fact_9234_ATP_Olambda__1054,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_ye(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_1054
tff(fact_9235_ATP_Olambda__1055,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_atk(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).

% ATP.lambda_1055
tff(fact_9236_ATP_Olambda__1056,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_rv(fun(C,A),fun(fun(D,B),fun(C,fun(D,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(D,B,Uua,Uuc)) ).

% ATP.lambda_1056
tff(fact_9237_ATP_Olambda__1057,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_ael(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).

% ATP.lambda_1057
tff(fact_9238_ATP_Olambda__1058,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_aax(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(B,bool,Uua,Uuc)) ) ) ).

% ATP.lambda_1058
tff(fact_9239_ATP_Olambda__1059,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_sb(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_1059
tff(fact_9240_ATP_Olambda__1060,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_kp(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uua,Uuc),aa(fun(C,bool),set(C),collect(C),aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_kl(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1060
tff(fact_9241_ATP_Olambda__1061,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(C),Uua: fun(B,fun(C,A)),Uub: fun(B,fun(C,bool)),Uuc: B] : aa(B,A,aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aTP_Lamp_km(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),Uu),Uua),Uub),Uuc) = groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uua,Uuc),aa(fun(C,bool),set(C),collect(C),aa(B,fun(C,bool),aa(fun(B,fun(C,bool)),fun(B,fun(C,bool)),aTP_Lamp_kl(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1061
tff(fact_9242_ATP_Olambda__1062,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: list(B),Uub: B,Uuc: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(list(B),fun(B,fun(list(B),list(B))),aTP_Lamp_pw(fun(B,A),fun(list(B),fun(B,fun(list(B),list(B)))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(B),product_prod(list(B),list(B))),list(B),aa(fun(list(B),fun(product_prod(list(B),list(B)),list(B))),fun(product_prod(list(B),product_prod(list(B),list(B))),list(B)),product_case_prod(list(B),product_prod(list(B),list(B)),list(B)),aTP_Lamp_pv(fun(B,A),fun(list(B),fun(product_prod(list(B),list(B)),list(B))),Uu)),linorder_part(B,A,Uu,aa(B,A,Uu,aa(nat,B,nth(B,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(B),nat,size_size(list(B)),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one))))),Uua)) ) ).

% ATP.lambda_1062
tff(fact_9243_ATP_Olambda__1063,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_aon(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
       => pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ) ).

% ATP.lambda_1063
tff(fact_9244_ATP_Olambda__1064,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_aqm(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_1064
tff(fact_9245_ATP_Olambda__1065,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_wd(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,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_wc(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_1065
tff(fact_9246_ATP_Olambda__1066,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,fun(A,bool)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_afp(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A9: A] :
            ( ( Uub = A9 )
            & ( Uuc = A9 ) )
        | ? [A9: A,B7: A,C5: A] :
            ( ( Uub = A9 )
            & ( Uuc = C5 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A9),B7))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B7),C5)) ) ) ) ).

% ATP.lambda_1066
tff(fact_9247_ATP_Olambda__1067,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,fun(A,bool)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_afq(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A9: A,B7: A] :
            ( ( Uub = A9 )
            & ( Uuc = B7 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,A9),B7)) )
        | ? [A9: A,B7: A,C5: A] :
            ( ( Uub = A9 )
            & ( Uuc = C5 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A9),B7))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B7),C5)) ) ) ) ).

% ATP.lambda_1067
tff(fact_9248_ATP_Olambda__1068,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(list(A),fun(list(A),bool)),Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_aft(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [Y4: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
        | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y4)) )
        | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y4))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,Y4),X4))
            & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uua,Xs3),Ys4)) ) ) ) ).

% ATP.lambda_1068
tff(fact_9249_ATP_Olambda__1069,axiom,
    ! [A: $tType,B: $tType,Uu: bool,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_cp(bool,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_1069
tff(fact_9250_ATP_Olambda__1070,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_nh(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_1070
tff(fact_9251_ATP_Olambda__1071,axiom,
    ! [A: $tType,B: $tType,Uu: bool,Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_em(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(Uu)
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_1071
tff(fact_9252_ATP_Olambda__1072,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(A,A),Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aov(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,aa(A,A,Uua,aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one)))) ) ).

% ATP.lambda_1072
tff(fact_9253_ATP_Olambda__1073,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aou(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),one_one(nat))) ) ).

% ATP.lambda_1073
tff(fact_9254_ATP_Olambda__1074,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_afd(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_1074
tff(fact_9255_ATP_Olambda__1075,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B,Uub: heap_ext(product_unit),Uuc: nat] : aa(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_ee(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_timeFrame(A,Uuc,aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,aa(B,heap_Time_Heap(A),Uu,Uua)),Uub)) ).

% ATP.lambda_1075
tff(fact_9256_ATP_Olambda__1076,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: A,Uub: heap_ext(product_unit),Uuc: nat] : aa(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_eh(fun(A,heap_Time_Heap(B)),fun(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_timeFrame(B,Uuc,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),Uu,Uua)),Uub)) ).

% ATP.lambda_1076
tff(fact_9257_ATP_Olambda__1077,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(bool),aa(A,fun(list(A),option(bool)),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_pr(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua),Uub),Uuc) = subset_eq_mset_impl(A,Uu,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),Uuc)) ).

% ATP.lambda_1077
tff(fact_9258_ATP_Olambda__1078,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_vr(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,A,Uua,aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_vp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1078
tff(fact_9259_ATP_Olambda__1079,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_vu(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,A,Uub,aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_vp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1079
tff(fact_9260_ATP_Olambda__1080,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_vq(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7311177749621191930dd_sum(B,A,Uua,aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_vp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1080
tff(fact_9261_ATP_Olambda__1081,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_vt(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = groups7311177749621191930dd_sum(B,A,Uub,aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_vp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1081
tff(fact_9262_ATP_Olambda__1082,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_aex(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),ord_min(A),Uu),min_list(A,Uua)) ) ).

% ATP.lambda_1082
tff(fact_9263_ATP_Olambda__1083,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(C,D),Uua: fun(A,fun(B,C)),Uub: A,Uuc: B] : aa(B,D,aa(A,fun(B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_dq(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),Uu),Uua),Uub),Uuc) = aa(C,D,Uu,aa(B,C,aa(A,fun(B,C),Uua,Uub),Uuc)) ).

% ATP.lambda_1083
tff(fact_9264_ATP_Olambda__1084,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,fun(list(A),B)),Uub: A,Uuc: list(A)] : aa(list(A),C,aa(A,fun(list(A),C),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_qa(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(list(A),B,aa(A,fun(list(A),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1084
tff(fact_9265_ATP_Olambda__1085,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_in(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(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_1085
tff(fact_9266_ATP_Olambda__1086,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_iy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(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_1086
tff(fact_9267_ATP_Olambda__1087,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_ir(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1087
tff(fact_9268_ATP_Olambda__1088,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_gp(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1088
tff(fact_9269_ATP_Olambda__1089,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: nat,Uub: list(A),Uuc: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_aar(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,take(A,Uua,Uub)),Uuc))) ) ).

% ATP.lambda_1089
tff(fact_9270_ATP_Olambda__1090,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_rr(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,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(D,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_1090
tff(fact_9271_ATP_Olambda__1091,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_rq(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,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(D,C,Uua,Uuc))) ).

% ATP.lambda_1091
tff(fact_9272_ATP_Olambda__1092,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: A,Uuc: B] : aa(B,fun(product_prod(C,D),bool),aa(A,fun(B,fun(product_prod(C,D),bool)),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool))),aTP_Lamp_auz(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),aa(B,fun(C,fun(D,bool)),aa(A,fun(B,fun(C,fun(D,bool))),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool)))),aTP_Lamp_auy(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1092
tff(fact_9273_ATP_Olambda__1093,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_pf(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),Uu),Uua),Uub),Uuc) = aa(fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_case_prod(list(B),list(B),product_prod(list(B),product_prod(list(B),list(B)))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_pe(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc)) ) ).

% ATP.lambda_1093
tff(fact_9274_ATP_Olambda__1094,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_ed(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ec(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1094
tff(fact_9275_ATP_Olambda__1095,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_eb(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ea(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1095
tff(fact_9276_ATP_Olambda__1096,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( semiring_0(B)
        & comm_monoid_add(A)
        & times(A) )
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: multiset(C),Uuc: A] : aa(A,B,aa(multiset(C),fun(A,B),aa(fun(C,B),fun(multiset(C),fun(A,B)),aTP_Lamp_aix(fun(A,B),fun(fun(C,B),fun(multiset(C),fun(A,B))),Uu),Uua),Uub),Uuc) = comm_m7189776963980413722m_mset(B,aa(multiset(C),multiset(B),image_mset(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_aiw(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1096
tff(fact_9277_ATP_Olambda__1097,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_yf(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),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ye(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1097
tff(fact_9278_ATP_Olambda__1098,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_xn(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)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_xm(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1098
tff(fact_9279_ATP_Olambda__1099,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_atl(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),image2(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_atk(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_1099
tff(fact_9280_ATP_Olambda__1100,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_zc(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_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_zb(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1100
tff(fact_9281_ATP_Olambda__1101,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_yq(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_yp(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1101
tff(fact_9282_ATP_Olambda__1102,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_oa(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),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_1102
tff(fact_9283_ATP_Olambda__1103,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_om(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_1103
tff(fact_9284_ATP_Olambda__1104,axiom,
    ! [A: $tType,Uu: fun(A,bool),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_uc(fun(A,bool),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,aa(fun(A,bool),fun(A,bool),comp(bool,bool,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_1104
tff(fact_9285_ATP_Olambda__1105,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_ph(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_1105
tff(fact_9286_ATP_Olambda__1106,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_ok(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_1106
tff(fact_9287_ATP_Olambda__1107,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_oi(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_1107
tff(fact_9288_ATP_Olambda__1108,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(C))),Uub: B,Uuc: A] : aa(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_ajz(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uua,Uuc)),aa(set(B),set(B),image(B,B,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uub),bot_bot(set(B)))))) ).

% ATP.lambda_1108
tff(fact_9289_ATP_Olambda__1109,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(set(product_prod(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_zq(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),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(D,E)),set(set(A)),image2(product_prod(D,E),set(A),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc))),Uua)) ).

% ATP.lambda_1109
tff(fact_9290_ATP_Olambda__1110,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aa(B,fun(fun(A,B),bool),aa(set(B),fun(B,fun(fun(A,B),bool)),aTP_Lamp_ale(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X4: A] :
          ( ( pp(aa(set(A),bool,member(A,X4),Uu))
           => pp(aa(set(B),bool,member(B,aa(A,B,Uuc,X4)),Uua)) )
          & ( ~ pp(aa(set(A),bool,member(A,X4),Uu))
           => ( aa(A,B,Uuc,X4) = Uub ) ) ) ) ).

% ATP.lambda_1110
tff(fact_9291_ATP_Olambda__1111,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_adt(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [Y4: C] :
          ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y4)),Uu))
          & pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y4),Uuc)),Uua)) ) ) ).

% ATP.lambda_1111
tff(fact_9292_ATP_Olambda__1112,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_aep(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A9: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A9)),aa(C,B,Uub,A9)) )
          & pp(aa(set(C),bool,member(C,A9),Uu)) ) ) ).

% ATP.lambda_1112
tff(fact_9293_ATP_Olambda__1113,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_acv(fun(B,A),fun(B,fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [L2: B] :
          ( ( Uuc = aa(B,A,Uu,L2) )
          & ( ( L2 = Uua )
            | pp(aa(set(B),bool,member(B,L2),Uub)) ) ) ) ).

% ATP.lambda_1113
tff(fact_9294_ATP_Olambda__1114,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A,Uub: fun(A,B),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ado(A,fun(A,fun(fun(A,B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ? [I: A] :
              ( ( Uuc = aa(A,B,Uub,I) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),I))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),Uua)) ) ) ) ).

% ATP.lambda_1114
tff(fact_9295_ATP_Olambda__1115,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,C)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(B,C)),fun(A,fun(A,bool)),aTP_Lamp_ami(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X4: product_prod(B,B)] :
          ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),X4),Uu))
         => pp(aa(product_prod(B,B),bool,aa(fun(B,fun(B,bool)),fun(product_prod(B,B),bool),product_case_prod(B,B,bool),aa(A,fun(B,fun(B,bool)),aa(A,fun(A,fun(B,fun(B,bool))),aTP_Lamp_amh(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,bool)))),Uua),Uub),Uuc)),X4)) ) ) ).

% ATP.lambda_1115
tff(fact_9296_ATP_Olambda__1116,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( pp(aa(C,bool,aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_adc(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [X4: A,Y4: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X4),Y4) )
          & pp(aa(A,bool,Uu,X4))
          & pp(aa(B,bool,Uua,Y4)) ) ) ).

% ATP.lambda_1116
tff(fact_9297_ATP_Olambda__1117,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_adn(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A19: A,A25: A] :
          ( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A19),A25) )
          & pp(aa(set(A),bool,member(A,A19),Uu))
          & pp(aa(set(A),bool,member(A,A25),Uu))
          & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A19)),aa(A,B,Uub,A25))),Uua)) ) ) ).

% ATP.lambda_1117
tff(fact_9298_ATP_Olambda__1118,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_us(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uua),Uub),aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1118
tff(fact_9299_ATP_Olambda__1119,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_un(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(A,C)),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uua),Uub),aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1119
tff(fact_9300_ATP_Olambda__1120,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: list(A),Uub: A,Uuc: list(A),Uud: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_ui(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),quicksort_by_rel(A,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),aa(list(A),list(A),quicksort_by_rel(A,Uu,Uua),Uud))),Uuc) ).

% ATP.lambda_1120
tff(fact_9301_ATP_Olambda__1121,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,Uu: fun(D,fun(E,C)),Uua: fun(A,D),Uub: fun(B,E),Uuc: A,Uud: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_aem(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(E,C,aa(D,fun(E,C),Uu,aa(A,D,Uua,Uuc)),aa(B,E,Uub,Uud)) ).

% ATP.lambda_1121
tff(fact_9302_ATP_Olambda__1122,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(E,C),Uuc: D,Uud: E] : aa(E,A,aa(D,fun(E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aek(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(E,C,Uub,Uud)) ).

% ATP.lambda_1122
tff(fact_9303_ATP_Olambda__1123,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(C,A)),Uua: fun(A,fun(B,fun(C,B))),Uub: A,Uuc: B,Uud: C] : aa(C,product_prod(A,B),aa(B,fun(C,product_prod(A,B)),aa(A,fun(B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_pn(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,aa(A,fun(C,A),Uu,Uub),Uud)),aa(C,B,aa(B,fun(C,B),aa(A,fun(B,fun(C,B)),Uua,Uub),Uuc),Uud)) ).

% ATP.lambda_1123
tff(fact_9304_ATP_Olambda__1124,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,C)),Uua: A,Uub: A,Uuc: B,Uud: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(A,fun(B,fun(B,bool)),aa(A,fun(A,fun(B,fun(B,bool))),aTP_Lamp_amh(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( aa(B,C,aa(A,fun(B,C),Uu,Uua),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uud) ) ) ).

% ATP.lambda_1124
tff(fact_9305_ATP_Olambda__1125,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_hq(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),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,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1125
tff(fact_9306_ATP_Olambda__1126,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_hn(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,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uuc),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1126
tff(fact_9307_ATP_Olambda__1127,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_ev(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,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 ) )
        & ( pp(aa(set(product_prod(A,A)),bool,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_1127
tff(fact_9308_ATP_Olambda__1128,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_ho(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1128
tff(fact_9309_ATP_Olambda__1129,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_aja(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uud),bot_bot(set(A))))))),field2(A,Uu)))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud))),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))),minus_minus(set(product_prod(A,A))),Uu),id2(A))))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc)),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))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = Uuc )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),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))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) ) ) ) ) ).

% ATP.lambda_1129
tff(fact_9310_ATP_Olambda__1130,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_ady(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(product_prod(A,A)),bool,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_rtrancl(A,Uub)))
        & pp(aa(set(product_prod(A,A)),bool,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_rtrancl(A,Uub))) ) ) ).

% ATP.lambda_1130
tff(fact_9311_ATP_Olambda__1131,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_nl(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),snga_assn(A,Uub,list_update(A,Uua,Uu,Uuc))),pure_assn(aa(array(A),bool,aa(array(A),fun(array(A),bool),fequal(array(A)),Uud),Uub))) ) ).

% ATP.lambda_1131
tff(fact_9312_ATP_Olambda__1132,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_aml(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( ( Uub = Uuc )
       => pp(aa(set(product_prod(A,A)),bool,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)),Uu)) ) ) ).

% ATP.lambda_1132
tff(fact_9313_ATP_Olambda__1133,axiom,
    ! [C: $tType,B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,fun(A,bool)),Uub: fun(A,fun(E,bool)),Uuc: C,Uud: E] :
      ( pp(aa(E,bool,aa(C,fun(E,bool),aa(fun(A,fun(E,bool)),fun(C,fun(E,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool))),aTP_Lamp_arj(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),Uu))))
          & pp(aa(A,bool,aa(C,fun(A,bool),Uua,Uuc),X4))
          & pp(aa(E,bool,aa(A,fun(E,bool),Uub,X4),Uud)) ) ) ).

% ATP.lambda_1133
tff(fact_9314_ATP_Olambda__1134,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
          ( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_aht(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ? [A9: A,B7: C] :
              ( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A9)),aa(C,B,Uua,B7)) )
              & pp(aa(set(A),bool,member(A,A9),Uub))
              & pp(aa(set(C),bool,member(C,B7),Uuc)) ) ) ) ).

% ATP.lambda_1134
tff(fact_9315_ATP_Olambda__1135,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B),Uud: list(B),Uue: list(B)] : aa(list(B),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_pe(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uuc)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),Uue)),if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub)),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uue))),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uud)),Uue)))) ) ).

% ATP.lambda_1135
tff(fact_9316_ATP_Olambda__1136,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: A,Uuc: B,Uud: C,Uue: D] :
      ( pp(aa(D,bool,aa(C,fun(D,bool),aa(B,fun(C,fun(D,bool)),aa(A,fun(B,fun(C,fun(D,bool))),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool)))),aTP_Lamp_auy(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),Uud))
        & pp(aa(D,bool,aa(B,fun(D,bool),Uua,Uuc),Uue)) ) ) ).

% ATP.lambda_1136
tff(fact_9317_ATP_Olambda__1137,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ec(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),Uu))
        | ( ( Uub = Uud )
          & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),Uua)) ) ) ) ).

% ATP.lambda_1137
tff(fact_9318_ATP_Olambda__1138,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ea(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( ( Uub = Uud )
        & pp(aa(A,bool,Uu,Uud))
        & pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),aa(A,set(product_prod(B,B)),Uua,Uud))) ) ) ).

% ATP.lambda_1138
tff(fact_9319_ATP_Olambda__1139,axiom,
    ! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_aai(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(Uu) ) ).

% ATP.lambda_1139
tff(fact_9320_ATP_Olambda__1140,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_bd(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).

% ATP.lambda_1140
tff(fact_9321_ATP_Olambda__1141,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(B)
     => ! [Uu: heap_Time_Heap(A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_bs(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1141
tff(fact_9322_ATP_Olambda__1142,axiom,
    ! [B: $tType,A: $tType,Uu: multiset(A),Uua: B] : aa(B,multiset(A),aTP_Lamp_yg(multiset(A),fun(B,multiset(A)),Uu),Uua) = Uu ).

% ATP.lambda_1142
tff(fact_9323_ATP_Olambda__1143,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_bj(assn,fun(A,assn),Uu),Uua) = Uu ).

% ATP.lambda_1143
tff(fact_9324_ATP_Olambda__1144,axiom,
    ! [A: $tType,Uu: bool,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_zu(bool,fun(A,bool),Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_1144
tff(fact_9325_ATP_Olambda__1145,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_abt(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1145
tff(fact_9326_ATP_Olambda__1146,axiom,
    ! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_aen(set(D),fun(B,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1146
tff(fact_9327_ATP_Olambda__1147,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_abq(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1147
tff(fact_9328_ATP_Olambda__1148,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_abr(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1148
tff(fact_9329_ATP_Olambda__1149,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_asd(set(B),fun(C,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1149
tff(fact_9330_ATP_Olambda__1150,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_ack(set(B),fun(B,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1150
tff(fact_9331_ATP_Olambda__1151,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_aaz(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1151
tff(fact_9332_ATP_Olambda__1152,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_acj(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1152
tff(fact_9333_ATP_Olambda__1153,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: C] : aa(C,set(A),aTP_Lamp_asc(set(A),fun(C,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1153
tff(fact_9334_ATP_Olambda__1154,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_ws(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1154
tff(fact_9335_ATP_Olambda__1155,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_abo(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1155
tff(fact_9336_ATP_Olambda__1156,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,fun(B,bool),aTP_Lamp_ati(fun(B,bool),fun(A,fun(B,bool)),Uu),Uua) = Uu ).

% ATP.lambda_1156
tff(fact_9337_ATP_Olambda__1157,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_nk(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).

% ATP.lambda_1157
tff(fact_9338_ATP_Olambda__1158,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_uw(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_1158
tff(fact_9339_ATP_Olambda__1159,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_rc(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).

% ATP.lambda_1159
tff(fact_9340_ATP_Olambda__1160,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ac(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).

% ATP.lambda_1160
tff(fact_9341_ATP_Olambda__1161,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_pz(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1161
tff(fact_9342_ATP_Olambda__1162,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_tr(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1162
tff(fact_9343_ATP_Olambda__1163,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ais(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1163
tff(fact_9344_ATP_Olambda__1164,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,B,aTP_Lamp_ce(B,fun(C,B),Uu),Uua) = Uu ).

% ATP.lambda_1164
tff(fact_9345_ATP_Olambda__1165,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_uf(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_1165
tff(fact_9346_ATP_Olambda__1166,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_wr(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1166
tff(fact_9347_ATP_Olambda__1167,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_wq(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1167
tff(fact_9348_ATP_Olambda__1168,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_afz(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1168
tff(fact_9349_ATP_Olambda__1169,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_mc(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1169
tff(fact_9350_ATP_Olambda__1170,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_qx(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1170
tff(fact_9351_ATP_Olambda__1171,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_agt(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1171
tff(fact_9352_ATP_Olambda__1172,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_md(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1172
tff(fact_9353_ATP_Olambda__1173,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_rl(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1173
tff(fact_9354_ATP_Olambda__1174,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_afi(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_1174
tff(fact_9355_ATP_Olambda__1175,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_sw(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_1175
tff(fact_9356_ATP_Olambda__1176,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_or(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_1176
tff(fact_9357_ATP_Olambda__1177,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_tk(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_1177
tff(fact_9358_ATP_Olambda__1178,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_ni(A,fun(B,B)),Uu),Uua) = Uua ).

% ATP.lambda_1178
tff(fact_9359_ATP_Olambda__1179,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_ql(A,fun(list(A),bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_1179
tff(fact_9360_ATP_Olambda__1180,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aur(A,fun(A,bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_1180
tff(fact_9361_ATP_Olambda__1181,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_qm(A,fun(list(A),bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1181
tff(fact_9362_ATP_Olambda__1182,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_en(A,fun(B,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1182
tff(fact_9363_ATP_Olambda__1183,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_dl(A,fun(A,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1183
tff(fact_9364_ATP_Olambda__1184,axiom,
    ! [Uu: product_prod(nat,nat)] : aa(product_prod(nat,nat),product_prod(nat,nat),aTP_Lamp_aut(product_prod(nat,nat),product_prod(nat,nat)),Uu) = Uu ).

% ATP.lambda_1184
tff(fact_9365_ATP_Olambda__1185,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_fn(nat,nat),Uu) = Uu ).

% ATP.lambda_1185
tff(fact_9366_ATP_Olambda__1186,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_ia(int,int),Uu) = Uu ).

% ATP.lambda_1186
tff(fact_9367_ATP_Olambda__1187,axiom,
    ! [D: $tType,Uu: fun(D,nat)] : aa(fun(D,nat),fun(D,nat),aTP_Lamp_anx(fun(D,nat),fun(D,nat)),Uu) = Uu ).

% ATP.lambda_1187
tff(fact_9368_ATP_Olambda__1188,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_aej(B,B),Uu) = Uu ).

% ATP.lambda_1188
tff(fact_9369_ATP_Olambda__1189,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ahh(A,A),Uu) = Uu ) ).

% ATP.lambda_1189
tff(fact_9370_ATP_Olambda__1190,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_tn(A,A),Uu) = Uu ) ).

% ATP.lambda_1190
tff(fact_9371_ATP_Olambda__1191,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wo(A,A),Uu) = Uu ) ).

% ATP.lambda_1191
tff(fact_9372_ATP_Olambda__1192,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_yh(A,A),Uu) = Uu ) ).

% ATP.lambda_1192
tff(fact_9373_ATP_Olambda__1193,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_pk(A,A),Uu) = Uu ) ).

% ATP.lambda_1193
tff(fact_9374_ATP_Olambda__1194,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ci(A,A),Uu) = Uu ) ).

% ATP.lambda_1194
tff(fact_9375_ATP_Olambda__1195,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_cc(A,A),Uu) = Uu ).

% ATP.lambda_1195
tff(fact_9376_ATP_Olambda__1196,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_zv(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_1196
tff(fact_9377_ATP_Olambda__1197,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(bool),aTP_Lamp_asn(A,set(bool)),Uu) = top_top(set(bool)) ).

% ATP.lambda_1197
tff(fact_9378_ATP_Olambda__1198,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_abb(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_1198
tff(fact_9379_ATP_Olambda__1199,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_vi(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1199
tff(fact_9380_ATP_Olambda__1200,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_xj(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_1200
tff(fact_9381_ATP_Olambda__1201,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_wp(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_1201
tff(fact_9382_ATP_Olambda__1202,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_vj(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_1202
tff(fact_9383_ATP_Olambda__1203,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_aba(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1203
tff(fact_9384_ATP_Olambda__1204,axiom,
    ! [B: $tType,Uu: B] : aa(B,nat,aTP_Lamp_agl(B,nat),Uu) = zero_zero(nat) ).

% ATP.lambda_1204
tff(fact_9385_ATP_Olambda__1205,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_fu(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1205
tff(fact_9386_ATP_Olambda__1206,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_tj(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1206
tff(fact_9387_ATP_Olambda__1207,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_cg(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1207
tff(fact_9388_ATP_Olambda__1208,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_akw(A,nat),Uu) = zero_zero(nat) ).

% ATP.lambda_1208
tff(fact_9389_ATP_Olambda__1209,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ch(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1209
tff(fact_9390_ATP_Olambda__1210,axiom,
    ! [Uu: product_unit] : aa(product_unit,assn,aTP_Lamp_bt(product_unit,assn),Uu) = one_one(assn) ).

% ATP.lambda_1210
tff(fact_9391_ATP_Olambda__1211,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ib(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1211
tff(fact_9392_ATP_Olambda__1212,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_mj(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_1212
tff(fact_9393_ATP_Olambda__1213,axiom,
    ! [A: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_az(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1213
tff(fact_9394_ATP_Olambda__1214,axiom,
    ! [B: $tType,D: $tType,Uu: B] : aa(B,option(D),aTP_Lamp_atx(B,option(D)),Uu) = none(D) ).

% ATP.lambda_1214
tff(fact_9395_ATP_Olambda__1215,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_nc(B,option(A)),Uu) = none(A) ).

% ATP.lambda_1215
tff(fact_9396_ATP_Olambda__1216,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_mt(A,option(C)),Uu) = none(C) ).

% ATP.lambda_1216
tff(fact_9397_ATP_Olambda__1217,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_bp(A,option(B)),Uu) = none(B) ).

% ATP.lambda_1217
tff(fact_9398_ATP_Olambda__1218,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_aij(A,B),Uu) = undefined(B) ).

% ATP.lambda_1218
tff(fact_9399_ATP_Olambda__1219,axiom,
    ! [Uu: product_prod(heap_ext(product_unit),set(nat))] :
      ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,aTP_Lamp_cx(product_prod(heap_ext(product_unit),set(nat)),bool),Uu))
    <=> $false ) ).

% ATP.lambda_1219
tff(fact_9400_ATP_Olambda__1220,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_fq(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1220
tff(fact_9401_ATP_Olambda__1221,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_aq(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1221
tff(fact_9402_ATP_Olambda__1222,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_fp(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1222
tff(fact_9403_ATP_Olambda__1223,axiom,
    ! [A: $tType,Uu: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aTP_Lamp_aqp(fun(A,bool),bool),Uu))
    <=> $true ) ).

% ATP.lambda_1223
tff(fact_9404_ATP_Olambda__1224,axiom,
    ! [D: $tType,Uu: D] :
      ( pp(aa(D,bool,aTP_Lamp_apd(D,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1224
tff(fact_9405_ATP_Olambda__1225,axiom,
    ! [C: $tType,Uu: C] :
      ( pp(aa(C,bool,aTP_Lamp_apf(C,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1225
tff(fact_9406_ATP_Olambda__1226,axiom,
    ! [B: $tType,Uu: B] :
      ( pp(aa(B,bool,aTP_Lamp_ave(B,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1226
tff(fact_9407_ATP_Olambda__1227,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_av(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1227
tff(fact_9408_ATP_Olambda__1228,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_att(B,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1228
tff(fact_9409_ATP_Olambda__1229,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_rb(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1229

% Type constructors (662)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_1,axiom,
    bounded_lattice(assn) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_2,axiom,
    ! [A20: $tType] :
      ( bounded_lattice_top(A20)
     => bounded_lattice(option(A20)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_3,axiom,
    ! [A20: $tType] : bounded_lattice(filter(A20)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_4,axiom,
    bounded_lattice(bool) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_5,axiom,
    ! [A20: $tType] : bounded_lattice(set(A20)) ).

tff(tcon_fun___Lattices_Obounded__lattice_6,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounded_lattice(fun(A20,A21)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple6319245703460814977attice(A21)
     => condit1219197933456340205attice(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple592849572758109894attice(A21)
     => comple592849572758109894attice(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple489889107523837845lgebra(A21)
     => comple489889107523837845lgebra(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounde4967611905675639751up_bot(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounde4346867609351753570nf_top(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple6319245703460814977attice(A21)
     => comple6319245703460814977attice(fun(A20,A21)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A20: $tType,A21: $tType] :
      ( boolea8198339166811842893lgebra(A21)
     => boolea8198339166811842893lgebra(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bounded_lattice(A21)
     => bounded_lattice_top(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A20: $tType,A21: $tType] :
      ( comple6319245703460814977attice(A21)
     => comple9053668089753744459l_ccpo(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_sup(A21)
     => semilattice_sup(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_inf(A21)
     => semilattice_inf(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( distrib_lattice(A21)
     => distrib_lattice(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A20: $tType,A21: $tType] :
      ( complete_Sup(A21)
     => complete_Sup(fun(A20,A21)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A20: $tType,A21: $tType] :
      ( complete_Inf(A21)
     => complete_Inf(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A20: $tType,A21: $tType] :
      ( order_top(A21)
     => order_top(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A20: $tType,A21: $tType] :
      ( order_bot(A21)
     => order_bot(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A20: $tType,A21: $tType] :
      ( preorder(A21)
     => preorder(fun(A20,A21)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( finite_finite(A20)
        & finite_finite(A21) )
     => finite_finite(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A20: $tType,A21: $tType] :
      ( lattice(A21)
     => lattice(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A20: $tType,A21: $tType] :
      ( order(A21)
     => order(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A20: $tType,A21: $tType] :
      ( top(A21)
     => top(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ord(A21)
     => ord(fun(A20,A21)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A20: $tType,A21: $tType] :
      ( bot(A21)
     => bot(fun(A20,A21)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A20: $tType,A21: $tType] :
      ( uminus(A21)
     => uminus(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Osup,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_sup(A21)
     => sup(fun(A20,A21)) ) ).

tff(tcon_fun___Lattices_Oinf,axiom,
    ! [A20: $tType,A21: $tType] :
      ( semilattice_inf(A21)
     => inf(fun(A20,A21)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_7,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___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___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(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___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___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___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_8,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_9,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_10,axiom,
    distrib_lattice(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___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___Complete__Lattices_OSup_11,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_12,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___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_13,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_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_14,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___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_15,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Orderings_Oord_16,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_17,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_18,axiom,
    sup(int) ).

tff(tcon_Int_Oint___Lattices_Oinf_19,axiom,
    inf(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(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___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___Heap_Oheap,axiom,
    heap(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_20,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_21,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_22,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_23,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_24,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_25,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_26,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_27,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_28,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_29,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_30,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_31,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_32,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_33,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_34,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_35,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_36,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_37,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_38,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_39,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_40,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_41,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_47,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_48,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_49,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_50,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_51,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_52,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_53,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_54,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_55,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_56,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_57,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_58,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_59,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_60,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_61,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_62,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_63,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_64,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_65,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_66,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_72,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_73,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_74,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_75,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_76,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_77,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_78,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_79,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_80,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_81,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_82,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_83,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_84,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_85,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_86,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_87,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_88,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_89,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_90,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Lattices_Osup_91,axiom,
    sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Oinf_92,axiom,
    inf(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_93,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Power_Opower_94,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_95,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_96,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_97,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_98,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_99,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_100,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_101,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_102,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_103,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_104,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_105,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_106,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_107,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_108,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_109,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_110,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_111,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_112,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_113,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_114,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_115,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_116,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_117,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_118,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_119,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_120,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_121,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_122,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_123,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_124,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_125,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_126,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_127,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_128,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_129,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_130,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_131,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_132,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_133,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_134,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_135,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_136,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_137,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_138,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_139,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_140,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_141,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_142,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_143,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_144,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_145,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_146,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_147,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_148,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_149,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_150,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_151,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_152,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_153,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_154,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_155,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_156,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_157,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_158,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_159,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_160,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_161,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_162,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_163,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_164,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_165,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_166,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_167,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_168,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_169,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_170,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_171,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_172,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_173,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_174,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_175,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_176,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_177,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Lattices_Osup_178,axiom,
    sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Oinf_179,axiom,
    inf(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_180,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_181,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_182,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_183,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_184,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_185,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_186,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_187,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_188,axiom,
    ! [A20: $tType] : condit1219197933456340205attice(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_189,axiom,
    ! [A20: $tType] : comple592849572758109894attice(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_190,axiom,
    ! [A20: $tType] : comple489889107523837845lgebra(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_191,axiom,
    ! [A20: $tType] : bounde4967611905675639751up_bot(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_192,axiom,
    ! [A20: $tType] : bounde4346867609351753570nf_top(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_193,axiom,
    ! [A20: $tType] : comple6319245703460814977attice(set(A20)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_194,axiom,
    ! [A20: $tType] : boolea8198339166811842893lgebra(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_195,axiom,
    ! [A20: $tType] : bounded_lattice_top(set(A20)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_196,axiom,
    ! [A20: $tType] : comple9053668089753744459l_ccpo(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_197,axiom,
    ! [A20: $tType] : semilattice_sup(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_198,axiom,
    ! [A20: $tType] : semilattice_inf(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_199,axiom,
    ! [A20: $tType] : distrib_lattice(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_200,axiom,
    ! [A20: $tType] : complete_Sup(set(A20)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_201,axiom,
    ! [A20: $tType] : complete_Inf(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_202,axiom,
    ! [A20: $tType] : order_top(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_203,axiom,
    ! [A20: $tType] : order_bot(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_204,axiom,
    ! [A20: $tType] : preorder(set(A20)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_205,axiom,
    ! [A20: $tType] :
      ( finite_finite(A20)
     => finite_finite(set(A20)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_206,axiom,
    ! [A20: $tType] : lattice(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oorder_207,axiom,
    ! [A20: $tType] : order(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Otop_208,axiom,
    ! [A20: $tType] : top(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Oord_209,axiom,
    ! [A20: $tType] : ord(set(A20)) ).

tff(tcon_Set_Oset___Orderings_Obot_210,axiom,
    ! [A20: $tType] : bot(set(A20)) ).

tff(tcon_Set_Oset___Groups_Ouminus_211,axiom,
    ! [A20: $tType] : uminus(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Osup_212,axiom,
    ! [A20: $tType] : sup(set(A20)) ).

tff(tcon_Set_Oset___Lattices_Oinf_213,axiom,
    ! [A20: $tType] : inf(set(A20)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_214,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_215,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_216,axiom,
    comple489889107523837845lgebra(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_217,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_218,axiom,
    bounde4346867609351753570nf_top(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_219,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_220,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_221,axiom,
    bounded_lattice_top(bool) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_222,axiom,
    comple9053668089753744459l_ccpo(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_223,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_224,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_225,axiom,
    distrib_lattice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_226,axiom,
    complete_Sup(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_227,axiom,
    complete_Inf(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_228,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_229,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_230,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_231,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_232,axiom,
    finite_finite(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_233,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_234,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Otop_235,axiom,
    top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_236,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Orderings_Obot_237,axiom,
    bot(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_238,axiom,
    uminus(bool) ).

tff(tcon_HOL_Obool___Lattices_Osup_239,axiom,
    sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Oinf_240,axiom,
    inf(bool) ).

tff(tcon_HOL_Obool___Heap_Oheap_241,axiom,
    heap(bool) ).

tff(tcon_Heap_Oref___Heap_Oheap_242,axiom,
    ! [A20: $tType] : heap(ref(A20)) ).

tff(tcon_Heap_Oref___Nat_Osize_243,axiom,
    ! [A20: $tType] : size(ref(A20)) ).

tff(tcon_List_Olist___Heap_Oheap_244,axiom,
    ! [A20: $tType] :
      ( heap(A20)
     => heap(list(A20)) ) ).

tff(tcon_List_Olist___Nat_Osize_245,axiom,
    ! [A20: $tType] : size(list(A20)) ).

tff(tcon_Heap_Oarray___Heap_Oheap_246,axiom,
    ! [A20: $tType] : heap(array(A20)) ).

tff(tcon_Heap_Oarray___Nat_Osize_247,axiom,
    ! [A20: $tType] : size(array(A20)) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_248,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Heap_Oheap_249,axiom,
    heap(char) ).

tff(tcon_String_Ochar___Nat_Osize_250,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_251,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( finite_finite(A20)
        & finite_finite(A21) )
     => finite_finite(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___Heap_Oheap_252,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( heap(A20)
        & heap(A21) )
     => heap(sum_sum(A20,A21)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_253,axiom,
    ! [A20: $tType,A21: $tType] : size(sum_sum(A20,A21)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_254,axiom,
    ! [A20: $tType] : condit1219197933456340205attice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_255,axiom,
    ! [A20: $tType] : bounde4967611905675639751up_bot(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_256,axiom,
    ! [A20: $tType] : bounde4346867609351753570nf_top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_257,axiom,
    ! [A20: $tType] : comple6319245703460814977attice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_258,axiom,
    ! [A20: $tType] : bounded_lattice_top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_259,axiom,
    ! [A20: $tType] : comple9053668089753744459l_ccpo(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_260,axiom,
    ! [A20: $tType] : semilattice_sup(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_261,axiom,
    ! [A20: $tType] : semilattice_inf(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_262,axiom,
    ! [A20: $tType] : distrib_lattice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_263,axiom,
    ! [A20: $tType] : complete_Sup(filter(A20)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_264,axiom,
    ! [A20: $tType] : complete_Inf(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_265,axiom,
    ! [A20: $tType] : order_top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_266,axiom,
    ! [A20: $tType] : order_bot(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_267,axiom,
    ! [A20: $tType] : preorder(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_268,axiom,
    ! [A20: $tType] : lattice(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_269,axiom,
    ! [A20: $tType] : order(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_270,axiom,
    ! [A20: $tType] : top(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_271,axiom,
    ! [A20: $tType] : ord(filter(A20)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_272,axiom,
    ! [A20: $tType] : bot(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Osup_273,axiom,
    ! [A20: $tType] : sup(filter(A20)) ).

tff(tcon_Filter_Ofilter___Lattices_Oinf_274,axiom,
    ! [A20: $tType] : inf(filter(A20)) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_275,axiom,
    ! [A20: $tType] :
      ( comple5582772986160207858norder(A20)
     => condit6923001295902523014norder(option(A20)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_276,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => condit1219197933456340205attice(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_277,axiom,
    ! [A20: $tType] :
      ( comple592849572758109894attice(A20)
     => comple592849572758109894attice(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_278,axiom,
    ! [A20: $tType] :
      ( lattice(A20)
     => bounde4967611905675639751up_bot(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_279,axiom,
    ! [A20: $tType] :
      ( bounded_lattice_top(A20)
     => bounde4346867609351753570nf_top(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
    ! [A20: $tType] :
      ( comple5582772986160207858norder(A20)
     => comple5582772986160207858norder(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_280,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple6319245703460814977attice(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_281,axiom,
    ! [A20: $tType] :
      ( bounded_lattice_top(A20)
     => bounded_lattice_top(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_282,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple9053668089753744459l_ccpo(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_283,axiom,
    ! [A20: $tType] :
      ( semilattice_sup(A20)
     => semilattice_sup(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_284,axiom,
    ! [A20: $tType] :
      ( semilattice_inf(A20)
     => semilattice_inf(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_285,axiom,
    ! [A20: $tType] :
      ( distrib_lattice(A20)
     => distrib_lattice(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OSup_286,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => complete_Sup(option(A20)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_OInf_287,axiom,
    ! [A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => complete_Inf(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_288,axiom,
    ! [A20: $tType] :
      ( wellorder(A20)
     => wellorder(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_289,axiom,
    ! [A20: $tType] :
      ( order_top(A20)
     => order_top(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_290,axiom,
    ! [A20: $tType] :
      ( order(A20)
     => order_bot(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_291,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => preorder(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_292,axiom,
    ! [A20: $tType] :
      ( linorder(A20)
     => linorder(option(A20)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_293,axiom,
    ! [A20: $tType] :
      ( finite_finite(A20)
     => finite_finite(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_294,axiom,
    ! [A20: $tType] :
      ( lattice(A20)
     => lattice(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_295,axiom,
    ! [A20: $tType] :
      ( order(A20)
     => order(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_296,axiom,
    ! [A20: $tType] :
      ( order_top(A20)
     => top(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_297,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => ord(option(A20)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_298,axiom,
    ! [A20: $tType] :
      ( order(A20)
     => bot(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Osup_299,axiom,
    ! [A20: $tType] :
      ( sup(A20)
     => sup(option(A20)) ) ).

tff(tcon_Option_Ooption___Lattices_Oinf_300,axiom,
    ! [A20: $tType] :
      ( inf(A20)
     => inf(option(A20)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_301,axiom,
    ! [A20: $tType] :
      ( heap(A20)
     => heap(option(A20)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_302,axiom,
    ! [A20: $tType] : size(option(A20)) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_303,axiom,
    bounde4967611905675639751up_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_304,axiom,
    bounde4346867609351753570nf_top(assn) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_305,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_306,axiom,
    bounded_lattice_top(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_307,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_308,axiom,
    semilattice_inf(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_309,axiom,
    distrib_lattice(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_310,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_311,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_312,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_313,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_314,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_315,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_316,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Otop_317,axiom,
    top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_318,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_319,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_320,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osup_321,axiom,
    sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Oinf_322,axiom,
    inf(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_323,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_324,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_325,axiom,
    dvd(assn) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_326,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => ordere6658533253407199908up_add(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_327,axiom,
    ! [A20: $tType] : cancel2418104881723323429up_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_328,axiom,
    ! [A20: $tType] : cancel1802427076303600483id_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_329,axiom,
    ! [A20: $tType] : cancel_semigroup_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_330,axiom,
    ! [A20: $tType] : comm_monoid_diff(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_331,axiom,
    ! [A20: $tType] : ab_semigroup_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_332,axiom,
    ! [A20: $tType] : comm_monoid_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OSup_333,axiom,
    ! [A20: $tType] : complete_Sup(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Complete__Lattices_OInf_334,axiom,
    ! [A20: $tType] : complete_Inf(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_335,axiom,
    ! [A20: $tType] : semigroup_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_336,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => preorder(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_337,axiom,
    ! [A20: $tType] : monoid_add(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_338,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => order(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_339,axiom,
    ! [A20: $tType] :
      ( preorder(A20)
     => ord(multiset(A20)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_340,axiom,
    ! [A20: $tType] : zero(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_341,axiom,
    ! [A20: $tType] : plus(multiset(A20)) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_342,axiom,
    ! [A20: $tType] : size(multiset(A20)) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_343,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( finite_finite(A20)
        & finite_finite(A21) )
     => finite_finite(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_344,axiom,
    ! [A20: $tType,A21: $tType] :
      ( ( heap(A20)
        & heap(A21) )
     => heap(product_prod(A20,A21)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_345,axiom,
    ! [A20: $tType,A21: $tType] : size(product_prod(A20,A21)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_346,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_347,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_348,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_349,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_350,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_351,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_352,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_353,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_354,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_355,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_356,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_357,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_358,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_359,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_360,axiom,
    complete_Sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_361,axiom,
    complete_Inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_362,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_363,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_364,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_365,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_366,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_367,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_368,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_369,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_370,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_371,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_372,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_373,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osup_374,axiom,
    sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Oinf_375,axiom,
    inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_376,axiom,
    heap(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_377,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_378,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_379,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_380,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_381,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_382,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_383,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_384,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_385,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_386,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_387,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_388,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_389,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_390,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_391,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_392,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_393,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_394,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_395,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_396,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_397,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_398,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_399,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_400,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_401,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_402,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_403,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_404,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_405,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_406,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_407,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_408,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_409,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_410,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_411,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_412,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_413,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_414,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_415,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_416,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_417,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_418,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_419,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_420,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_421,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_422,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_423,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_424,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_425,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_426,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_427,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_428,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_429,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_430,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_431,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_432,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_433,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_434,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_435,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_436,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_437,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_438,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_439,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_440,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_441,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_442,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_443,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_444,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_445,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_446,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_447,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_448,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_449,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_450,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_451,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_452,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_453,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_454,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_455,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_456,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_457,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_458,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_459,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_460,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_461,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_462,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_463,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_464,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_465,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_466,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_467,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_468,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_469,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_470,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_471,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_472,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_473,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_474,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_475,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_476,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_477,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_478,axiom,
    linord4140545234300271783up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_479,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_480,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_481,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_482,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_483,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_484,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_485,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_486,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_487,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_488,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_489,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_490,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_491,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_492,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_493,axiom,
    semiring_1_cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_494,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_495,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_496,axiom,
    comm_monoid_diff(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_497,axiom,
    ab_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_498,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_499,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_500,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_501,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_502,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_503,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_504,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_505,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_506,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_507,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_508,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_509,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_510,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_511,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_512,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_513,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_514,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_515,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_516,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_517,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_518,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_519,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_520,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_521,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Otimes_522,axiom,
    times(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_523,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_524,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_525,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oplus_526,axiom,
    plus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_527,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_528,axiom,
    size(code_natural) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_529,axiom,
    ! [A20: $tType] : size(heap_Time_Heap(A20)) ).

% Helper facts (17)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A] :
      ( ~ pp(aa(fun(A,bool),bool,fAll(A),P2))
      | pp(aa(A,bool,P2,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P2: bool] :
      ( pp(P2)
      | pp(aa(bool,bool,fNot,P2)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P2: bool] :
      ( ~ pp(aa(bool,bool,fNot,P2))
      | ~ pp(P2) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P2: bool,Q2: bool] :
      ( ~ pp(fconj(P2,Q2))
      | pp(Q2) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P2: bool,Q2: bool] :
      ( ~ pp(fconj(P2,Q2))
      | pp(P2) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P2: bool,Q2: bool] :
      ( ~ pp(P2)
      | ~ pp(Q2)
      | pp(fconj(P2,Q2)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P2: bool,Q2: bool] :
      ( ~ pp(fdisj(P2,Q2))
      | pp(P2)
      | pp(Q2) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q2: bool,P2: bool] :
      ( ~ pp(Q2)
      | pp(fdisj(P2,Q2)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P2: bool,Q2: bool] :
      ( ~ pp(P2)
      | pp(fdisj(P2,Q2)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P2: bool] :
      ( ( P2 = fTrue )
      | ( P2 = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P2: fun(A,bool)] : aa(A,bool,P2,fChoice(A,P2)) = aa(fun(A,bool),bool,fEx(A),P2) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,rep_assn(aa(b,assn,q,rg)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),h),hoare_new_addrs(h3,as,h)))) ).

%------------------------------------------------------------------------------